title: Victoria (Australia)

text: Melbourne has the world's largest tram network, currently operated by Yarra Trams. As well as being a popular form of public transport, over the last few decades trams have become one of Melbourne's major tourist attractions. There are also tourist trams operating over portions of the former Ballarat and Bendigo systems. There are also tramway museums at Bylands and Haddon.


title: Victoria (Australia)

text: Melbourne Airport is the major domestic and international gateway for the state. Avalon Airport is the state's second busiest airport, which complements Essendon and Moorabbin Airports to see the remainder of Melbourne's air traffic. Hamilton Airport, Mildura Airport, Mount Hotham and Portland Airport are the remaining airports with scheduled domestic flights. There are no fewer than 27 other airports in the state with no scheduled flights.


title: Victoria (Australia)

text: The Port of Melbourne is the largest port for containerised and general cargo in Australia, and is located in Melbourne on the mouth of the Yarra River, which is at the head of Port Phillip. Additional seaports are at Westernport, Geelong, and Portland.


title: Victoria (Australia)

text: As of October 2013, smoking tobacco is prohibited in the sheltered areas of train stations, and tram and bus stops—between 2012 and 2013, 2002 people were issued with infringement notices. The state government announced a plan in October 2013 to prohibit smoking on all Victorian railway station platforms and raised tram stops.


title: Victoria (Australia)

text: Victoria's major utilities include a collection of brown-coal-fired power stations, particularly in the Latrobe Valley. One of these was the recently decommissioned Hazelwood Power Station, which was number 1 on the worldwide List of least carbon efficient power stations. The Victorian government is aiming to cut 40.6 mega tonnes of greenhouse gases emissions by 2025.


title: Victoria (Australia)

text: Victoria's water infrastructure includes a series of dams and reservoirs, predominantly in Central Victoria, that hold and collect water for much of the state. The water collected is of a very high quality and requires little chlorination treatment, giving the water a taste more like water collected in a rainwater tank. In regional areas however, such as in the west of the state, chlorination levels are much higher.


title: Victoria (Australia)

text: The Victorian Water Grid consists of a number of new connections and pipelines being built across the State. This allows water to be moved around Victoria to where it is needed most and reduces the impact of localised droughts in an era thought to be influenced by climate change. Major projects already completed as part of the Grid include the Wimmera Mallee Pipeline and the Goldfields Superpipe.


title: Victoria (Australia)

text: Victoria is the home of Australian rules football, with ten of the 18 Australian Football League (AFL) clubs based in the state. The AFL Grand Final is traditionally held at the Melbourne Cricket Ground on the last Saturday of September. The state has a public holiday the day before the Grand Final, which coincides with the AFL Grand Final parade.


title: Victoria (Australia)

text: The Victorian cricket team play in the national Sheffield Shield cricket competition. Victoria is represented in the National Rugby League by the Melbourne Storm and in Super Rugby by the Melbourne Rebels. It is represented in the National Basketball League by Melbourne United and South East Melbourne Phoenix. It is also represented in soccer by Melbourne Victory, Melbourne City and Western United in the A-League.


title: Victoria (Australia)

text: Melbourne has held the 1956 Summer Olympics, 2006 Commonwealth Games and the FINA World Swimming Championship.


title: Victoria (Australia)

text: Melbourne is also home to the Australian Open tennis tournament in January each year, the first of the world's four Grand Slam tennis tournaments, and the Australian Formula One Australian Grand Prix in March. It hosted the Australian Masters golf tournament from 1979 to 2015.


title: Victoria (Australia)

text: Victoria's Bells Beach hosts one of the world's longest-running surfing competition, the Bells Beach SurfClassic, which is part of The ASP World Tour.


title: Victoria (Australia)

text: The Melbourne Vixens and Collingwood Magpies Netball represent Victoria in the National Netball League.


title: Victoria (Australia)

text: Victoria's Phillip Island is home of the Phillip Island Grand Prix Circuit which hosts the Australian motorcycle Grand Prix which features MotoGP (the world's premier motorcycling class), as well as the Australian round of the World Superbike Championship and the domestic V8 Supercar racing, which also visits Sandown Raceway and the rural Winton Motor Raceway circuit.


title: Victoria (Australia)

text: Victoria is also home to the Aussie Millions poker tournament, the richest in the Southern Hemisphere.


title: Victoria (Australia)

text: The main horse racing tracks in Victoria are Caulfield Racecourse, Flemington Racecourse and Sandown Racecourse. The Melbourne Spring Racing Carnival is one of the biggest horse racing events in the world and is one of the world's largest sporting events. The main race is for the $6 million Melbourne Cup, and crowds for the carnival usually exceed 700,000.


title: Prashanth Neel

text: Prashanth Neelakantapuram (born 4 June 1980) is an Indian film director and screenwriter who primarily works in the Kannada film industry. He made his debut with the 2014 Kannada film, "Ugramm" and later directed the two-part film series: "" (2018), and "" (2022), the latter being the highest-grossing Kannada film.


title: Prashanth Neel

text: Prashanth Neelakantapuram was born on 4 June 1980 in a Telugu family in Karnataka. Born to Subhash and Bharathi, their family hails from Neelakantapuram village, near Madakasira, Andhra Pradesh, and settled in Bangalore.


title: Prashanth Neel

text: Neel married Likhitha in 2010. The couple has a daughter and son. Neel's sister is married to Kannada actor Sriimurali, who eventually starred in Neel's directorial debut "Ugramm". Neel is also a cousin of Telugu film actor, Aadarsh Balakrishna, and is also related to former Andhra Pradesh state minister, Raghu Veera Reddy.


title: Prashanth Neel

text: In an interview with a Kannada weekly magazine, Neel said he took to filmmaking initially for the need of money, following which he completed a course in the same, eventually developing interest in it. After having completed the course, he decided to cast his brother-in-law and actor Srimurali in " Aa Hudugi Neene", a screenplay that he had written during the time. After the latter expressed his concerns due to his inexperience, Neel changed his mind and began observing Srimurali's mannerisms over a period of time, and came up with the project "Ugramm", an action film. The film turned out to be a massive commercial success and emerged as one of the highest-grossing Kannada films of 2014.


title: Prashanth Neel

text: Reports emerged in late 2014 and early 2015 of Neel having been signed to direct three films. The first is a sequel to "Ugramm", entitled "Ugramm Veeram", which would feature Srimurali reprise his role from "Ugramm". His next film "KGF" which stands for Kolar Gold Fields presented Yash in the role of Rocky, an orphan who becomes a don. The sequel "", was scheduled for a theatrical release on 16 July 2021, but was delayed due to the second wave of the COVID pandemic in India.


title: Prashanth Neel

text: On 2 December 2020, Hombale Films announced the film "Salaar" with Prabhas. The film's shoot began on 29 January 2021. In May 2021, Mythri Movie Makers and N.T.R. Arts have announced a film in collaboration with Neel and Jr. NTR, for his 31st film. The film is expected to begin its production in April 2023.


title: Baywatch (film)

text: Baywatch is a 2017 American action comedy film directed by Seth Gordon, with a screenplay by Mark Swift and Damian Shannon, from a story by Jay Scherick, David Ronn, Thomas Lennon, and Robert Ben Garant. It is based on the television series created by Michael Berk, Douglas Schwartz, and Gregory J. Bonann. The film stars Dwayne Johnson and Zac Efron. The story follows lifeguard Mitch Buchannon and his team who must take down a drug lord in an effort to save their beach.


title: Baywatch (film)

text: Principal photography began on February 22, 2016, in Florida and Savannah, Georgia. The film was released in the United States on May 25, 2017, by Paramount Pictures, grossed $177 million worldwide and received negative reviews. "Baywatch" received five nominations at the 38th Golden Raspberry Awards, including Worst Picture and Worst Actor for Efron.


title: Baywatch (film)

text: In Emerald Bay, Florida, Lt. Mitch Buchannon and his team of lifeguards, including second-in-command Stephanie Holden and veteran C.J. Parker, protect the beaches and the bay as part of an elite division known as Baywatch.


title: Baywatch (film)

text: With over 500 rescues in his career, Mitch is beloved by the community, annoying beat cop Garner Ellerbee and his superior, Captain Thorpe. On morning patrol, Mitch finds a small pouch of flakka by the Huntley Club, now owned by businesswoman Victoria Leeds, who has been bribing city councilman Rodriguez and others to control the bay.


title: Baywatch (film)

text: Mitch later meets Matt Brody, who thinks being an Olympic gold medal-winning swimmer entitles him to be on the team without trying out. He and Holden tease him, Mitch later makes Brody try to beat him on the "big boys'" course. As the course is ending, a mom and her kids start drowning, so the lifeguards and Brody rush in to save them, although Brody isn't a lifeguard yet. Finally, those who made the tryouts are Ronnie Greenbaum, Summer Quinn, and Brody.


title: Baywatch (film)

text: When a private yacht catches fire, the occupants are evacuated, except for deceased councilman Rodriguez. Ellerbee disapproves of Mitch's further investigative actions. After Mitch and the gang attend Leeds' party without finding out about the flakka, Mitch, Brody, and Summer later infiltrate the hospital morgue. Videoing Leeds' henchmen planting an autopsy report to cover up Rodriguez's murder, they are unfortunately discovered, the recording is destroyed and Ellerbee releases their suspect on a lack of evidence. Angered, Thorpe threatens to fire Mitch if he oversteps again.


title: Baywatch (film)

text: Convinced Leeds is running drugs through the Huntley, Mitch and Brody go undercover. In the kitchen, they witness the drugs being retrieved from barrels of fish. Contacting Ellerbee, they learn another dead body has been found on the beach. Thorpe, enraged that Mitch essentially deserted his post, fires him, naming Brody the new lieutenant. Brody doesn't want the job but is forced to take it and Mitch takes a phone store job.


title: Baywatch (film)

text: Brody later sees "sand grifters (beach thieves)" steal bags using a cooler. He orders them off the beach, and keeps the cooler. After finding another pouch of flakka on the beach, Brody takes the second victim's report from Ellerbee using the cooler and brings it to Summer, who confirms the man was killed by knife instead of a shark attack. Ronnie, recognizing the victim as his friend Dave, realizes that Ronnie had told Brody that Dave was working the Leeds for the Huntley. Brody realizes that Mitch was right about the Huntley, before Ronnie helps Brody hack into Leeds' servers, and finds her plan to privatize the entire beach by eliminating the competition.


title: Baywatch (film)

text: The team, infiltrating a private party on Leeds' personal yacht, discover that she's been using the hull to smuggle the drugs. Captured, Brody is tied up and put in a bait cage, and a gloating Leeds reveals she bribed Thorpe to replace Mitch with Brody before pushing the cage into the water. Before Brody can drown, Summer swims to him to give him the kiss of life. However, he's hallucinating and realizes it is actually Mitch.


title: Baywatch (film)

text: They catch up to Leeds, while Ronnie and C. J. launch fireworks at her escape helicopter. As they run out, a wounded Mitch, having intentionally stung himself with a sea urchin to boost his adrenaline, uses a Roman candle to blow Leeds up. Ellerbee arrives, takes her henchmen into custody, and apologizes to Mitch for doubting him. Thorpe arrives, berating Mitch for returning to the beach despite being fired. In response, Brody punches him in the face and Thorpe is subsequently arrested for his role in Leeds' plan.


title: Baywatch (film)

text: Sometime later, Ronnie and Brody begin relationships with C.J. and Summer respectively. Mitch, reinstated, announces that Summer, Ronnie, and Brody are no longer trainees and introduces them to their new captain, Casey Jean.


title: Baywatch (film)

text: A "Baywatch" movie was first announced in 2004, although the movie became stuck in development hell over the years, with several writers penning drafts. In July 2015, Sean Anders was replaced by Seth Gordon as director. On October 2, 2014, Dwayne Johnson was attached to star in the lead role, and Justin Malen was set to rewrite the script. Damian Shannon and Mark Swift wrote the latest draft, and the film would be comedic in style. On August 10, 2015, Zac Efron signed on to star in the film, and Beau Flynn and Ivan Reitman joined to produce with Johnson's Seven Bucks Productions. On November 9, 2015, "Deadline" reported that seven actresses were among the short list testing for the lead female role, Alexandra Daddario, Ashley Benson, Nina Dobrev, Alexandra Shipp, Shelley Hennig, Bianca A. Santos, and Denyse Tontz. On November 18, 2015, Johnson confirmed Daddario would play Summer, a lifeguard, and the love interest of Efron's character.


title: Baywatch (film)

text: On January 4, 2016, Kelly Rohrbach's casting as C. J. Parker was confirmed by Johnson's Instagram post. Damian Shannon and Mark Swift wrote the screenplay, while the final list of producers were Johnson along with his partner Dany Garcia, through their Seven Bucks Productions, as well as Flynn, Reitman, Michael Berk, Douglas Schwartz, and Gregory J. Bonann. On January 20, 2016, Johnson again posted on his Instagram about casting Ilfenesh Hadera as Johnson's love interest. On January 27, 2016, Jon Bass was cast in the film to play Ronnie, a funny, awkward, and skilled disco dancer at the beach, who falls in love with Parker. On February 12, 2016, Hannibal Buress joined the cast of the film to play a bay community local. On February 17, 2016, Priyanka Chopra signed on to star as the antagonist in the film. The role was originally written for a male. In March 2016, Yahya Abdul-Mateen II joined the cast as Ellerbee, a police officer who is constantly reminding Mitch that he has no real authority over the bay. Jack Kesy and Amin Joseph were also cast in the film. Vine (later YouTube) star Logan Paul announced that he is in the film, but his scenes were later cut. Izabel Goulart also appeared. NFL players Vernon Davis and Arian Foster also have cameo appearances in the film.


title: Baywatch (film)

text: Principal photography on the film began on February 22, 2016, in Deerfield Beach, Florida, with the setting in Broward County, Florida, while the TV series was set in Malibu, California. The film was shot in Miami and Savannah, Georgia. In late March 2016, filming started in Tybee Island, Georgia. Filming wrapped on May 18, 2016.


title: Baywatch (film)

text: In January 2016, Paramount Pictures scheduled "Baywatch" for a May 19, 2017 release, which was originally scheduled for a sequel to "Terminator Genisys". In December 2016, Paramount pushed the release date back one week to May 26, 2017, to avoid competition with "". In April 2017, the film was moved one day earlier, to avoid direct competition with "".


title: Baywatch (film)

text: "Baywatch" grossed $58.1 million in the United States and Canada and $119.8 million in other territories, for a worldwide total of $177.8 million, against a production budget of $69 million (not including marketing and distribution costs).


title: Baywatch (film)

text: In the United States and Canada, the film was initially projected to gross around $40 million from 3,642 theaters over its five-day opening weekend, with the studio predicting a more conservative $30 million debut. It made $1.25 million at 2,554 theaters from Wednesday night previews, similar to fellow mid-week R-rated releases "We're the Millers" ($1.7 million) and "Let's Be Cops" ($1.2 million). However, after making $4.6 million on its opening day (including Wednesday night previews) and $5.7 million on Friday, five-day projections were lowered to $25 million. It ended up grossing $18.5 million in its opening weekend (for a five-day total of $27.7 million), finishing third at the box office, behind "" ($63 million) and "Guardians of the Galaxy Vol. 2" ($20.9 million). In its second weekend the film grossed $8.8 million (a drop of 52.5%), finishing 5th at the box office.


title: Baywatch (film)

text: On review aggregation website Rotten Tomatoes, the film has an approval rating of 17% based on 246 reviews, and an average rating of 4.00/10. The website's critical consensus reads, ""Baywatch" takes its source material's jiggle factor to R-rated levels, but lacks the original's campy charm – and leaves its charming stars flailing in the shallows." On Metacritic the film has a weighted average score of 37 out of 100, based on 47 critics, indicating "generally unfavorable reviews". Audiences polled by CinemaScore gave the film an average grade of "B+" on an A+ to F scale, while PostTrak reported filmgoers gave it a 76% overall positive score. Dwayne Johnson responded to critics' reviews, saying that the film was made for fans and that they loved it.


title: Baywatch (film)

text: Writing for "Rolling Stone", Peter Travers praised the "easy rapport" of Johnson and Efron while saying, "what [the film] needs more is a functional script". Travers rated it two out of four stars and said, "Think of yourself sitting down for a big two-hour wallow in instant stupid with a vat of popcorn, slathered in fake butter and possibly a mound of melted M&Ms on top. It feels great chugging it down, then your stomach hurts, your head aches and you puke the whole thing up so you can forget about it in the morning. That's "Baywatch" in a nutshell." Owen Gleiberman of "Variety" called it "stupidly entertaining... for a while" but was critical of the plot. Brian Truitt of "USA Today" gave the film two out of four stars, and said that while the film has all the elements you would expect "the remake yearns to be both sendup farce and straight action film, tripping along the way and failing to grasp either." Richard Roeper of the "Chicago Sun-Times" gave the film 1.5 out of 4 stars, writing: "As was the case with "CHiPS", "The Dukes of Hazzard", "The Beverly Hillbillies", "Car 54, Where Are You?" and I'll just stop there, when you make films from junk TV, more often than not you're going to wind up with a junk movie."


title: Baywatch (film)

text: In May 2017, producer Beau Flynn revealed plans for a sequel were in development, with the project's story already written. Damian Shannon and Mark Swift were stated to be returning to their roles as co-screenwriters, while the plot would revolve around the titular team's adventures in traveling to a different environment overseas. Dwayne Johnson and Zac Efron entered early negotiations to reprise their roles, while the remaining cast were expected to also return in the potential sequel.


title: Lewis Pullman

text: Lewis James Pullman (born January 29, 1993) is an American actor. His film credits include "", "Bad Times at the El Royale" (both 2018), and "" (2022). On television, he starred as Major Major Major Major in "Catch-22" and Rhett Abbott in "Outer Range". His upcoming projects include the lead role in the film adaptation of Stephen King's "Salem's Lot".


title: Lewis Pullman

text: Lewis James Pullman was born on January 29, 1993, in Los Angeles, California. He is the son of actor Bill Pullman and modern dancer Tamara Hurwitz. He has a sister, Maesa, who is a singer-songwriter, and a brother, Jack, who is a puppet-maker. His mother is of Norwegian and Jewish descent.


title: Lewis Pullman

text: Pullman plays the drums in the band Atta Boy alongside Eden Brolin, Freddy Reish, and Dashel Thompson. They released their first album, "Out of Sorts", in 2012 as a "whimsical experiment." Their second album, "Big Heart Manners", was released in 2020 after an eight-year hiatus.


title: Lewis Pullman

text: Pullman began his acting career with several short films, starting with "The Tutor" in 2013. He had to split his time between living in Los Angeles and Montana. He said he was "on the tractor division [in college]. I figured if acting didn't work out I could be on the road crew, working the back hoe." In 2015, he graduated from Warren Wilson College in Swannanoa, North Carolina, with a bachelor's degree in Social Work. That same year, he was asked by filmmakers Jonathan Dayton and Valerie Faris to audition for "Highston", a television series from Sacha Baron Cohen and Amazon Studios. In September 2015, his casting was confirmed and the series was given a six-episode order. The series starred Pullman in the lead role as a 19-year-old whose imaginary friends were real-life celebrities. The pilot episode, guest-starring Flea and Shaquille O'Neal, was well received by critics. In December 2017, however, "Highston" was canceled after only one episode.


title: Lewis Pullman

text: Pullman made his feature film acting debut in 2017 with "The Ballad of Lefty Brown", a Western starring his father in the titular lead role. His additional roles in 2017 include the Arnold Schwarzenegger-led "Aftermath", the British drama "Lean on Pete", and the critically acclaimed film "Battle of the Sexes", the latter also starring his father.


title: Lewis Pullman

text: In 2018, Pullman had a leading role in the slasher film "". Though the film received mixed reviews when compared to its predecessor, it was a box-office success, grossing $32.1 million against a $5 million production budget. That same year, he starred alongside Jeff Bridges, Cynthia Erivo, Dakota Johnson, Jon Hamm, Cailee Spaeny, and Chris Hemsworth in "Bad Times at the El Royale". On casting Pullman, writer and director Drew Goddard said "it was one of those good old-fashioned casting searches. After meeting with lots and lots and lots of actors, Lewis came in and you just felt that immediately. The last time that happened, quite honestly, was when Chris Hemsworth walked in for "The Cabin in the Woods". You're just looking for actors who inherently fit the role — and then also transcend the role. Lewis had that sort of magic." Upon release, the film received generally positive reviews from critics, and his performance was also singled out, with "Den of Geek" calling it a "standout" and "The Seattle Times" writing that he "shines as the troubled desk clerk, who serves as the film's (very) late-arriving moral conscience."


title: Lewis Pullman

text: In 2019, Pullman had a recurring role as Major Major Major Major in "Catch-22", the Hulu adaptation of the 1961 novel of the same name starring George Clooney. That same year, he starred in the thriller film "Them That Follow". In 2022, Pullman played Rhett Abbott in the Amazon series "Outer Range", Lieutenant Robert "Bob" Floyd in the Tom Cruise-led sequel "", and the romantic lead in "Press Play". He will next star in the James Wan-produced horror film "Salem's Lot", an adaptation of the 1975 novel of the same name by Stephen King; "The Hollywood Reporter" writes that the role "promises to be a breakthrough and career-making for Pullman."


title: Lewis Pullman

text: In August 2022, it was revealed Pullman would star alongside Brie Larson in the Apple TV+ series "Lessons in Chemistry".


title: Eigenvalues and eigenvectors

text: In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by formula_1, is the factor by which the eigenvector is scaled.


title: Eigenvalues and eigenvectors

text: Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated.


title: Eigenvalues and eigenvectors

text: If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as


title: Eigenvalues and eigenvectors

text: where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root associated with .


title: Eigenvalues and eigenvectors

text: There is a direct correspondence between "n"-by-"n" square matrices and linear transformations from an "n"-dimensional vector space into itself, given any basis of the vector space. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices, or the language of linear transformations.


title: Eigenvalues and eigenvectors

text: Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix "eigen-" is adopted from the German word "eigen" (cognate with the English word "own") for 'proper', 'characteristic', 'own'. Originally used to study principal axes of the rotational motion of rigid bodies, eigenvalues and eigenvectors have a wide range of applications, for example in stability analysis, vibration analysis, atomic orbitals, facial recognition, and matrix diagonalization.


title: Eigenvalues and eigenvectors

text: In essence, an eigenvector v of a linear transformation "T" is a nonzero vector that, when "T" is applied to it, does not change direction. Applying "T" to the eigenvector only scales the eigenvector by the scalar value "λ", called an eigenvalue. This condition can be written as the equation


title: Eigenvalues and eigenvectors

text: referred to as the eigenvalue equation or eigenequation. In general, "λ" may be any scalar. For example, "λ" may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or complex.


title: Eigenvalues and eigenvectors

text: The Mona Lisa example pictured here provides a simple illustration. Each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example is called a shear mapping. Points in the top half are moved to the right, and points in the bottom half are moved to the left, proportional to how far they are from the horizontal axis that goes through the middle of the painting. The vectors pointing to each point in the original image are therefore tilted right or left, and made longer or shorter by the transformation. Points "along" the horizontal axis do not move at all when this transformation is applied. Therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation, because the mapping does not change its direction. Moreover, these eigenvectors all have an eigenvalue equal to one, because the mapping does not change their length either.


title: Eigenvalues and eigenvectors

text: Linear transformations can take many different forms, mapping vectors in a variety of vector spaces, so the eigenvectors can also take many forms. For example, the linear transformation could be a differential operator like formula_5, in which case the eigenvectors are functions called eigenfunctions that are scaled by that differential operator, such as


title: Eigenvalues and eigenvectors

text: Alternatively, the linear transformation could take the form of an "n" by "n" matrix, in which case the eigenvectors are "n" by 1 matrices. If the linear transformation is expressed in the form of an "n" by "n" matrix "A", then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication


title: Eigenvalues and eigenvectors

text: where the eigenvector "v" is an "n" by 1 matrix. For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix—for example by diagonalizing it.


title: Eigenvalues and eigenvectors

text: Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix "eigen-" is applied liberally when naming them:


title: Eigenvalues and eigenvectors

text: Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms and differential equations.


title: Eigenvalues and eigenvectors

text: In the 18th century, Leonhard Euler studied the rotational motion of a rigid body, and discovered the importance of the principal axes. Joseph-Louis Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.


title: Eigenvalues and eigenvectors

text: In the early 19th century, Augustin-Louis Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions. Cauchy also coined the term "racine caractéristique" (characteristic root), for what is now called "eigenvalue"; his term survives in "characteristic equation".


title: Eigenvalues and eigenvectors

text: Later, Joseph Fourier used the work of Lagrange and Pierre-Simon Laplace to solve the heat equation by separation of variables in his famous 1822 book "Théorie analytique de la chaleur". Charles-François Sturm developed Fourier's ideas further, and brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that real symmetric matrices have real eigenvalues. This was extended by Charles Hermite in 1855 to what are now called Hermitian matrices.


title: Eigenvalues and eigenvectors

text: Around the same time, Francesco Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle, and Alfred Clebsch found the corresponding result for skew-symmetric matrices. Finally, Karl Weierstrass clarified an important aspect in the stability theory started by Laplace, by realizing that defective matrices can cause instability.


title: Eigenvalues and eigenvectors

text: In the meantime, Joseph Liouville studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called "Sturm–Liouville theory". Schwarz studied the first eigenvalue of Laplace's equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's equation a few years later.


title: Eigenvalues and eigenvectors

text: At the start of the 20th century, David Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices. He was the first to use the German word "eigen", which means "own", to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Hermann von Helmholtz. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is the standard today.


title: Eigenvalues and eigenvectors

text: The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when Richard von Mises published the power method. One of the most popular methods today, the QR algorithm, was proposed independently by John G. F. Francis and Vera Kublanovskaya in 1961.


title: Eigenvalues and eigenvectors

text: Eigenvalues and eigenvectors are often introduced to students in the context of linear algebra courses focused on matrices.


title: Eigenvalues and eigenvectors

text: Furthermore, linear transformations over a finite-dimensional vector space can be represented using matrices, which is especially common in numerical and computational applications.


title: Eigenvalues and eigenvectors

text: Consider -dimensional vectors that are formed as a list of scalars, such as the three-dimensional vectors


title: Eigenvalues and eigenvectors

text: These vectors are said to be scalar multiples of each other, or parallel or collinear, if there is a scalar such that


title: Eigenvalues and eigenvectors

text: then is an eigenvector of the linear transformation and the scale factor is the eigenvalue corresponding to that eigenvector. Equation () is the eigenvalue equation for the matrix .


title: Eigenvalues and eigenvectors

text: Equation () has a nonzero solution "v" if and only if the determinant of the matrix is zero. Therefore, the eigenvalues of "A" are values of "λ" that satisfy the equation


title: Eigenvalues and eigenvectors

text: Using the Leibniz formula for determinants, the left-hand side of Equation () is a polynomial function of the variable "λ" and the degree of this polynomial is "n", the order of the matrix "A". Its coefficients depend on the entries of "A", except that its term of degree "n" is always (−1)"λ". This polynomial is called the "characteristic polynomial" of "A". Equation () is called the "characteristic equation" or the "secular equation" of "A".


title: Eigenvalues and eigenvectors

text: The fundamental theorem of algebra implies that the characteristic polynomial of an "n"-by-"n" matrix "A", being a polynomial of degree "n", can be factored into the product of "n" linear terms,


title: Eigenvalues and eigenvectors

text: where each "λ" may be real but in general is a complex number. The numbers "λ", "λ", ..., "λ", which may not all have distinct values, are roots of the polynomial and are the eigenvalues of "A".


title: Eigenvalues and eigenvectors

text: As a brief example, which is described in more detail in the examples section later, consider the matrix


title: Eigenvalues and eigenvectors

text: Setting the characteristic polynomial equal to zero, it has roots at and , which are the two eigenvalues of "A". The eigenvectors corresponding to each eigenvalue can be found by solving for the components of v in the equation In this example, the eigenvectors are any nonzero scalar multiples of


title: Eigenvalues and eigenvectors

text: If the entries of the matrix "A" are all real numbers, then the coefficients of the characteristic polynomial will also be real numbers, but the eigenvalues may still have nonzero imaginary parts. The entries of the corresponding eigenvectors therefore may also have nonzero imaginary parts. Similarly, the eigenvalues may be irrational numbers even if all the entries of "A" are rational numbers or even if they are all integers. However, if the entries of "A" are all algebraic numbers, which include the rationals, the eigenvalues are complex algebraic numbers.


title: Eigenvalues and eigenvectors

text: The non-real roots of a real polynomial with real coefficients can be grouped into pairs of complex conjugates, namely with the two members of each pair having imaginary parts that differ only in sign and the same real part. If the degree is odd, then by the intermediate value theorem at least one of the roots is real. Therefore, any real matrix with odd order has at least one real eigenvalue, whereas a real matrix with even order may not have any real eigenvalues. The eigenvectors associated with these complex eigenvalues are also complex and also appear in complex conjugate pairs.


title: Eigenvalues and eigenvectors

text: Let "λ" be an eigenvalue of an "n" by "n" matrix "A". The algebraic multiplicity "μ"("λ") of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is, the largest integer "k" such that ("λ" − "λ") divides evenly that polynomial.


title: Eigenvalues and eigenvectors

text: Suppose a matrix "A" has dimension "n" and "d" ≤ "n" distinct eigenvalues. Whereas Equation () factors the characteristic polynomial of "A" into the product of "n" linear terms with some terms potentially repeating, the characteristic polynomial can instead be written as the product of "d" terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity,


title: Eigenvalues and eigenvectors

text: If "d" = "n" then the right-hand side is the product of "n" linear terms and this is the same as Equation (). The size of each eigenvalue's algebraic multiplicity is related to the dimension "n" as


title: Eigenvalues and eigenvectors

text: If "μ"("λ") = 1, then "λ" is said to be a "simple eigenvalue". If "μ"("λ") equals the geometric multiplicity of "λ", "γ"("λ"), defined in the next section, then "λ" is said to be a "semisimple eigenvalue".


title: Eigenvalues and eigenvectors

text: Given a particular eigenvalue "λ" of the "n" by "n" matrix "A", define the set "E" to be all vectors v that satisfy Equation (),


title: Eigenvalues and eigenvectors

text: On one hand, this set is precisely the kernel or nullspace of the matrix ("A" − "λI"). On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of "A" associated with "λ". So, the set "E" is the union of the zero vector with the set of all eigenvectors of "A" associated with "λ", and "E" equals the nullspace of ("A" − "λI"). "E" is called the eigenspace or characteristic space of "A" associated with "λ". In general "λ" is a complex number and the eigenvectors are complex "n" by 1 matrices. A property of the nullspace is that it is a linear subspace, so "E" is a linear subspace of formula_20.


title: Eigenvalues and eigenvectors

text: Because the eigenspace "E" is a linear subspace, it is closed under addition. That is, if two vectors u and v belong to the set "E", written , then or equivalently . This can be checked using the distributive property of matrix multiplication. Similarly, because "E" is a linear subspace, it is closed under scalar multiplication. That is, if and "α" is a complex number, or equivalently . This can be checked by noting that multiplication of complex matrices by complex numbers is commutative. As long as u + v and "α"v are not zero, they are also eigenvectors of "A" associated with "λ".


title: Eigenvalues and eigenvectors

text: The dimension of the eigenspace "E" associated with "λ", or equivalently the maximum number of linearly independent eigenvectors associated with "λ", is referred to as the eigenvalue's geometric multiplicity "γ"("λ"). Because "E" is also the nullspace of ("A" − "λI"), the geometric multiplicity of "λ" is the dimension of the nullspace of ("A" − "λI"), also called the "nullity" of ("A" − "λI"), which relates to the dimension and rank of ("A" − "λI") as


title: Eigenvalues and eigenvectors

text: Because of the definition of eigenvalues and eigenvectors, an eigenvalue's geometric multiplicity must be at least one, that is, each eigenvalue has at least one associated eigenvector. Furthermore, an eigenvalue's geometric multiplicity cannot exceed its algebraic multiplicity. Additionally, recall that an eigenvalue's algebraic multiplicity cannot exceed "n".


title: Eigenvalues and eigenvectors

text: To prove the inequality formula_23, consider how the definition of geometric multiplicity implies the existence of formula_24 orthonormal eigenvectors formula_25, such that formula_26. We can therefore find a (unitary) matrix formula_27 whose first formula_24 columns are these eigenvectors, and whose remaining columns can be any orthonormal set of formula_29 vectors orthogonal to these eigenvectors of formula_30. Then formula_27 has full rank and is therefore invertible, and formula_32 with formula_33 a matrix whose top left block is the diagonal matrix formula_34. This implies that formula_35. In other words, formula_36 is similar to formula_37, which implies that formula_38. But from the definition of formula_33 we know that formula_40 contains a factor formula_41, which means that the algebraic multiplicity of formula_1 must satisfy formula_43.


title: Eigenvalues and eigenvectors

text: Suppose formula_30 has formula_45 distinct eigenvalues formula_46, where the geometric multiplicity of formula_47 is formula_48. The total geometric multiplicity of formula_30,


title: Eigenvalues and eigenvectors

text: is the dimension of the sum of all the eigenspaces of formula_30's eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of formula_30. If formula_53, then


title: Eigenvalues and eigenvectors

text: Let formula_30 be an arbitrary formula_62 matrix of complex numbers with eigenvalues formula_63. Each eigenvalue appears formula_64 times in this list, where formula_64 is the eigenvalue's algebraic multiplicity. The following are properties of this matrix and its eigenvalues:


title: Eigenvalues and eigenvectors

text: Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. For that reason, the word "eigenvector" in the context of matrices almost always refers to a right eigenvector, namely a "column" vector that "right" multiplies the formula_62 matrix formula_30 in the defining equation, Equation (),


title: Eigenvalues and eigenvectors

text: The eigenvalue and eigenvector problem can also be defined for "row" vectors that "left" multiply matrix formula_30. In this formulation, the defining equation is


title: Eigenvalues and eigenvectors

text: where formula_102 is a scalar and formula_103 is a formula_104 matrix. Any row vector formula_103 satisfying this equation is called a left eigenvector of formula_30 and formula_102 is its associated eigenvalue. Taking the transpose of this equation,


title: Eigenvalues and eigenvectors

text: Comparing this equation to Equation (), it follows immediately that a left eigenvector of formula_30 is the same as the transpose of a right eigenvector of formula_110, with the same eigenvalue. Furthermore, since the characteristic polynomial of formula_110 is the same as the characteristic polynomial of formula_30, the eigenvalues of the left eigenvectors of formula_30 are the same as the eigenvalues of the right eigenvectors of formula_110.


title: Eigenvalues and eigenvectors

text: Suppose the eigenvectors of "A" form a basis, or equivalently "A" has "n" linearly independent eigenvectors v, v, ..., v with associated eigenvalues "λ", "λ", ..., "λ". The eigenvalues need not be distinct. Define a square matrix "Q" whose columns are the "n" linearly independent eigenvectors of "A",


title: Eigenvalues and eigenvectors

text: Since each column of "Q" is an eigenvector of "A", right multiplying "A" by "Q" scales each column of "Q" by its associated eigenvalue,


title: Eigenvalues and eigenvectors

text: With this in mind, define a diagonal matrix Λ where each diagonal element Λ is the eigenvalue associated with the "i"th column of "Q". Then


title: Eigenvalues and eigenvectors

text: Because the columns of "Q" are linearly independent, Q is invertible. Right multiplying both sides of the equation by "Q",


title: Eigenvalues and eigenvectors

text: "A" can therefore be decomposed into a matrix composed of its eigenvectors, a diagonal matrix with its eigenvalues along the diagonal, and the inverse of the matrix of eigenvectors. This is called the eigendecomposition and it is a similarity transformation. Such a matrix "A" is said to be "similar" to the diagonal matrix Λ or "diagonalizable". The matrix "Q" is the change of basis matrix of the similarity transformation. Essentially, the matrices "A" and Λ represent the same linear transformation expressed in two different bases. The eigenvectors are used as the basis when representing the linear transformation as Λ.


title: Eigenvalues and eigenvectors

text: Conversely, suppose a matrix "A" is diagonalizable. Let "P" be a non-singular square matrix such that "P""AP" is some diagonal matrix "D". Left multiplying both by "P", . Each column of "P" must therefore be an eigenvector of "A" whose eigenvalue is the corresponding diagonal element of "D". Since the columns of "P" must be linearly independent for "P" to be invertible, there exist "n" linearly independent eigenvectors of "A". It then follows that the eigenvectors of "A" form a basis if and only if "A" is diagonalizable.


title: Eigenvalues and eigenvectors

text: A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvectors generalizes to generalized eigenvectors and the diagonal matrix of eigenvalues generalizes to the Jordan normal form. Over an algebraically closed field, any matrix "A" has a Jordan normal form and therefore admits a basis of generalized eigenvectors and a decomposition into generalized eigenspaces.


title: Eigenvalues and eigenvectors

text: In the Hermitian case, eigenvalues can be given a variational characterization. The largest eigenvalue of formula_120 is the maximum value of the quadratic form formula_121. A value of formula_122 that realizes that maximum, is an eigenvector.


title: Eigenvalues and eigenvectors

text: The figure on the right shows the effect of this transformation on point coordinates in the plane. The eigenvectors "v" of this transformation satisfy Equation (), and the values of "λ" for which the determinant of the matrix ("A" − "λI") equals zero are the eigenvalues.


title: Eigenvalues and eigenvectors

text: Setting the characteristic polynomial equal to zero, it has roots at and , which are the two eigenvalues of "A".


title: Eigenvalues and eigenvectors

text: Thus, the vectors v and v are eigenvectors of "A" associated with the eigenvalues and , respectively.


title: Eigenvalues and eigenvectors

text: The roots of the characteristic polynomial are 2, 1, and 11, which are the only three eigenvalues of "A". These eigenvalues correspond to the eigenvectors and or any nonzero multiple thereof.


title: Eigenvalues and eigenvectors

text: This matrix shifts the coordinates of the vector up by one position and moves the first coordinate to the bottom. Its characteristic polynomial is 1 − "λ", whose roots are


title: Eigenvalues and eigenvectors

text: For the real eigenvalue "λ" = 1, any vector with three equal nonzero entries is an eigenvector. For example,


title: Eigenvalues and eigenvectors

text: Therefore, the other two eigenvectors of "A" are complex and are formula_139 and formula_140 with eigenvalues "λ" and "λ", respectively. The two complex eigenvectors also appear in a complex conjugate pair,


title: Eigenvalues and eigenvectors

text: Matrices with entries only along the main diagonal are called "diagonal matrices". The eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider the matrix


title: Eigenvalues and eigenvectors

text: which has the roots , , and . These roots are the diagonal elements as well as the eigenvalues of "A".


title: Eigenvalues and eigenvectors

text: Each diagonal element corresponds to an eigenvector whose only nonzero component is in the same row as that diagonal element. In the example, the eigenvalues correspond to the eigenvectors,


title: Eigenvalues and eigenvectors

text: A matrix whose elements above the main diagonal are all zero is called a "lower triangular matrix", while a matrix whose elements below the main diagonal are all zero is called an "upper triangular matrix". As with diagonal matrices, the eigenvalues of triangular matrices are the elements of the main diagonal.


title: Eigenvalues and eigenvectors

text: The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The "algebraic multiplicity" of each eigenvalue is 2; in other words they are both double roots. The sum of the algebraic multiplicities of all distinct eigenvalues is "μ" = 4 = "n", the order of the characteristic polynomial and the dimension of "A".


title: Eigenvalues and eigenvectors

text: On the other hand, the "geometric multiplicity" of the eigenvalue 2 is only 1, because its eigenspace is spanned by just one vector formula_150 and is therefore 1-dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by just one vector formula_151. The total geometric multiplicity "γ" is 2, which is the smallest it could be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section.


title: Eigenvalues and eigenvectors

text: For a Hermitian matrix, the norm squared of the "j"th component of a normalized eigenvector can be calculated using only the matrix eigenvalues and the eigenvalues of the corresponding minor matrix,


title: Eigenvalues and eigenvectors

text: where formula_153 is the submatrix formed by removing the "j"th row and column from the original matrix. This identity also extends to diagonalizable matrices, and has been rediscovered many times in the literature.


title: Eigenvalues and eigenvectors

text: The definitions of eigenvalue and eigenvectors of a linear transformation "T" remains valid even if the underlying vector space is an infinite-dimensional Hilbert or Banach space. A widely used class of linear transformations acting on infinite-dimensional spaces are the differential operators on function spaces. Let "D" be a linear differential operator on the space C of infinitely differentiable real functions of a real argument "t". The eigenvalue equation for "D" is the differential equation


title: Eigenvalues and eigenvectors

text: The functions that satisfy this equation are eigenvectors of "D" and are commonly called eigenfunctions.


title: Eigenvalues and eigenvectors

text: This differential equation can be solved by multiplying both sides by "dt"/"f"("t") and integrating. Its solution, the exponential function


title: Eigenvalues and eigenvectors

text: is the eigenfunction of the derivative operator. In this case the eigenfunction is itself a function of its associated eigenvalue. In particular, for "λ" = 0 the eigenfunction "f"("t") is a constant.


title: Eigenvalues and eigenvectors

text: The concept of eigenvalues and eigenvectors extends naturally to arbitrary linear transformations on arbitrary vector spaces. Let "V" be any vector space over some field "K" of scalars, and let "T" be a linear transformation mapping "V" into "V",


title: Eigenvalues and eigenvectors

text: We say that a nonzero vector v ∈ "V" is an eigenvector of "T" if and only if there exists a scalar "λ" ∈ "K" such that


title: Eigenvalues and eigenvectors

text: This equation is called the eigenvalue equation for "T", and the scalar "λ" is the eigenvalue of "T" corresponding to the eigenvector v. "T"(v) is the result of applying the transformation "T" to the vector v, while "λ"v is the product of the scalar "λ" with v.


title: Eigenvalues and eigenvectors

text: which is the union of the zero vector with the set of all eigenvectors associated with "λ". "E" is called the eigenspace or characteristic space of "T" associated with "λ".


title: Eigenvalues and eigenvectors

text: for x, y ∈ "V" and "α" ∈ "K". Therefore, if u and v are eigenvectors of "T" associated with eigenvalue "λ", namely u, v ∈ "E", then


title: Eigenvalues and eigenvectors

text: So, both u + v and αv are either zero or eigenvectors of "T" associated with "λ", namely u + v, "α"v ∈ "E", and "E" is closed under addition and scalar multiplication. The eigenspace "E" associated with "λ" is therefore a linear subspace of "V".


title: Eigenvalues and eigenvectors

text: The geometric multiplicity "γ"("λ") of an eigenvalue "λ" is the dimension of the eigenspace associated with "λ", i.e., the maximum number of linearly independent eigenvectors associated with that eigenvalue. By the definition of eigenvalues and eigenvectors, "γ"("λ") ≥ 1 because every eigenvalue has at least one eigenvector.


title: Eigenvalues and eigenvectors

text: The eigenspaces of "T" always form a direct sum. As a consequence, eigenvectors of "different" eigenvalues are always linearly independent. Therefore, the sum of the dimensions of the eigenspaces cannot exceed the dimension "n" of the vector space on which "T" operates, and there cannot be more than "n" distinct eigenvalues.


title: Eigenvalues and eigenvectors

text: Any subspace spanned by eigenvectors of "T" is an invariant subspace of "T", and the restriction of "T" to such a subspace is diagonalizable. Moreover, if the entire vector space "V" can be spanned by the eigenvectors of "T", or equivalently if the direct sum of the eigenspaces associated with all the eigenvalues of "T" is the entire vector space "V", then a basis of "V" called an eigenbasis can be formed from linearly independent eigenvectors of "T". When "T" admits an eigenbasis, "T" is diagonalizable.


title: Eigenvalues and eigenvectors

text: If "λ" is an eigenvalue of "T", then the operator ("T" − "λI") is not one-to-one, and therefore its inverse ("T" − "λI") does not exist. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional vector spaces. In general, the operator ("T" − "λI") may not have an inverse even if "λ" is not an eigenvalue.


title: Eigenvalues and eigenvectors

text: For this reason, in functional analysis eigenvalues can be generalized to the spectrum of a linear operator "T" as the set of all scalars "λ" for which the operator ("T" − "λI") has no bounded inverse. The spectrum of an operator always contains all its eigenvalues but is not limited to them.


title: Eigenvalues and eigenvectors

text: One can generalize the algebraic object that is acting on the vector space, replacing a single operator acting on a vector space with an algebra representation – an associative algebra acting on a module. The study of such actions is the field of representation theory.


title: Eigenvalues and eigenvectors

text: The representation-theoretical concept of weight is an analog of eigenvalues, while "weight vectors" and "weight spaces" are the analogs of eigenvectors and eigenspaces, respectively.


title: Eigenvalues and eigenvectors

text: which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the "k" – 1 equations formula_164 giving a "k"-dimensional system of the first order in the stacked variable vector formula_165 in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This equation gives "k" characteristic roots formula_166 for use in the solution equation


title: Eigenvalues and eigenvectors

text: The calculation of eigenvalues and eigenvectors is a topic where theory, as presented in elementary linear algebra textbooks, is often very far from practice.


title: Eigenvalues and eigenvectors

text: The classical method is to first find the eigenvalues, and then calculate the eigenvectors for each eigenvalue. It is in several ways poorly suited for non-exact arithmetics such as floating-point.


title: Eigenvalues and eigenvectors

text: The eigenvalues of a matrix formula_30 can be determined by finding the roots of the characteristic polynomial. This is easy for formula_170 matrices, but the difficulty increases rapidly with the size of the matrix.


title: Eigenvalues and eigenvectors

text: In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of products of matrix elements; and there are algorithms that can find all the roots of a polynomial of arbitrary degree to any required accuracy. However, this approach is not viable in practice because the coefficients would be contaminated by unavoidable round-off errors, and the roots of a polynomial can be an extremely sensitive function of the coefficients (as exemplified by Wilkinson's polynomial). Even for matrices whose elements are integers the calculation becomes nontrivial, because the sums are very long; the constant term is the determinant, which for an formula_171 matrix is a sum of formula_172 different products.


title: Eigenvalues and eigenvectors

text: Explicit algebraic formulas for the roots of a polynomial exist only if the degree formula_57 is 4 or less. According to the Abel–Ruffini theorem there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree formula_57 is the characteristic polynomial of some companion matrix of order formula_57.) Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be computed by approximate numerical methods. Even the exact formula for the roots of a degree 3 polynomial is numerically impractical.


title: Eigenvalues and eigenvectors

text: Once the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be found by finding nonzero solutions of the eigenvalue equation, that becomes a system of linear equations with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix


title: Eigenvalues and eigenvectors

text: Both equations reduce to the single linear equation formula_181. Therefore, any vector of the form formula_182, for any nonzero real number formula_183, is an eigenvector of formula_30 with eigenvalue formula_185.


title: Eigenvalues and eigenvectors

text: The matrix formula_30 above has another eigenvalue formula_187. A similar calculation shows that the corresponding eigenvectors are the nonzero solutions of formula_188, that is, any vector of the form formula_189, for any nonzero real number formula_190.


title: Eigenvalues and eigenvectors

text: The converse approach, of first seeking the eigenvectors and then determining each eigenvalue from its eigenvector, turns out to be far more tractable for computers. The easiest algorithm here consists of picking an arbitrary starting vector and then repeatedly multiplying it with the matrix (optionally normalizing the vector to keep its elements of reasonable size); this makes the vector converge towards an eigenvector. A variation is to instead multiply the vector by this causes it to converge to an eigenvector of the eigenvalue closest to


title: Eigenvalues and eigenvectors

text: If formula_191 is (a good approximation of) an eigenvector of formula_30, then the corresponding eigenvalue can be computed as


title: Eigenvalues and eigenvectors

text: Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the QR algorithm was designed in 1961. Combining the Householder transformation with the LU decomposition results in an algorithm with better convergence than the QR algorithm. For large Hermitian sparse matrices, the Lanczos algorithm is one example of an efficient iterative method to compute eigenvalues and eigenvectors, among several other possibilities.


title: Eigenvalues and eigenvectors

text: Most numeric methods that compute the eigenvalues of a matrix also determine a set of corresponding eigenvectors as a by-product of the computation, although sometimes implementors choose to discard the eigenvector information as soon as it is no longer needed.


title: Eigenvalues and eigenvectors

text: The following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues, and eigenvectors.


title: Eigenvalues and eigenvectors

text: The characteristic equation for a rotation is a quadratic equation with discriminant formula_196, which is a negative number whenever is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers, formula_197; and all eigenvectors have non-real entries. Indeed, except for those special cases, a rotation changes the direction of every nonzero vector in the plane.


title: Eigenvalues and eigenvectors

text: A linear transformation that takes a square to a rectangle of the same area (a squeeze mapping) has reciprocal eigenvalues.


title: Eigenvalues and eigenvectors

text: An example of an eigenvalue equation where the transformation formula_198 is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics:


title: Eigenvalues and eigenvectors

text: where formula_120, the Hamiltonian, is a second-order differential operator and formula_201, the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue formula_202, interpreted as its energy.


title: Eigenvalues and eigenvectors

text: However, in the case where one is interested only in the bound state solutions of the Schrödinger equation, one looks for formula_201 within the space of square integrable functions. Since this space is a Hilbert space with a well-defined scalar product, one can introduce a basis set in which formula_201 and formula_120 can be represented as a one-dimensional array (i.e., a vector) and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form.


title: Eigenvalues and eigenvectors

text: The bra–ket notation is often used in this context. A vector, which represents a state of the system, in the Hilbert space of square integrable functions is represented by formula_206. In this notation, the Schrödinger equation is:


title: Eigenvalues and eigenvectors

text: where formula_206 is an eigenstate of formula_120 and formula_202 represents the eigenvalue. formula_120 is an observable self-adjoint operator, the infinite-dimensional analog of Hermitian matrices. As in the matrix case, in the equation above formula_212 is understood to be the vector obtained by application of the transformation formula_120 to formula_206.


title: Eigenvalues and eigenvectors

text: Light, acoustic waves, and microwaves are randomly scattered numerous times when traversing a static disordered system. Even though multiple scattering repeatedly randomizes the waves, ultimately coherent wave transport through the system is a deterministic process which can be described by a field transmission matrix formula_215. The eigenvectors of the transmission operator formula_216 form a set of disorder-specific input wavefronts which enable waves to couple into the disordered system's eigenchannels: the independent pathways waves can travel through the system. The eigenvalues, formula_217, of formula_216 correspond to the intensity transmittance associated with each eigenchannel. One of the remarkable properties of the transmission operator of diffusive systems is their bimodal eigenvalue distribution with formula_219 and formula_220. Furthermore, one of the striking properties of open eigenchannels, beyond the perfect transmittance, is the statistically robust spatial profile of the eigenchannels.


title: Eigenvalues and eigenvectors

text: In quantum mechanics, and in particular in atomic and molecular physics, within the Hartree–Fock theory, the atomic and molecular orbitals can be defined by the eigenvectors of the Fock operator. The corresponding eigenvalues are interpreted as ionization potentials via Koopmans' theorem. In this case, the term eigenvector is used in a somewhat more general meaning, since the Fock operator is explicitly dependent on the orbitals and their eigenvalues. Thus, if one wants to underline this aspect, one speaks of nonlinear eigenvalue problems. Such equations are usually solved by an iteration procedure, called in this case self-consistent field method. In quantum chemistry, one often represents the Hartree–Fock equation in a non-orthogonal basis set. This particular representation is a generalized eigenvalue problem called Roothaan equations.


title: Eigenvalues and eigenvectors

text: In geology, especially in the study of glacial till, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers. In the field, a geologist may collect such data for hundreds or thousands of clasts in a soil sample, which can only be compared graphically such as in a Tri-Plot (Sneed and Folk) diagram, or as a Stereonet on a Wulff Net.


title: Eigenvalues and eigenvectors

text: The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. The three eigenvectors are ordered formula_221 by their eigenvalues formula_222;


title: Eigenvalues and eigenvectors

text: formula_223 then is the primary orientation/dip of clast, formula_224 is the secondary and formula_225 is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on a compass rose of 360°. Dip is measured as the eigenvalue, the modulus of the tensor: this is valued from 0° (no dip) to 90° (vertical). The relative values of formula_226, formula_227, and formula_228 are dictated by the nature of the sediment's fabric. If formula_229, the fabric is said to be isotropic. If formula_230, the fabric is said to be planar. If formula_231, the fabric is said to be linear.


title: Eigenvalues and eigenvectors

text: The eigendecomposition of a symmetric positive semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used in multivariate analysis, where the sample covariance matrices are PSD. This orthogonal decomposition is called principal component analysis (PCA) in statistics. PCA studies linear relations among variables. PCA is performed on the covariance matrix or the correlation matrix (in which each variable is scaled to have its sample variance equal to one). For the covariance or correlation matrix, the eigenvectors correspond to principal components and the eigenvalues to the variance explained by the principal components. Principal component analysis of the correlation matrix provides an orthogonal basis for the space of the observed data: In this basis, the largest eigenvalues correspond to the principal components that are associated with most of the covariability among a number of observed data.


title: Eigenvalues and eigenvectors

text: Principal component analysis is used as a means of dimensionality reduction in the study of large data sets, such as those encountered in bioinformatics. In Q methodology, the eigenvalues of the correlation matrix determine the Q-methodologist's judgment of "practical" significance (which differs from the statistical significance of hypothesis testing; cf. criteria for determining the number of factors). More generally, principal component analysis can be used as a method of factor analysis in structural equation modeling.


title: Eigenvalues and eigenvectors

text: Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are the natural frequencies (or eigenfrequencies) of vibration, and the eigenvectors are the shapes of these vibrational modes. In particular, undamped vibration is governed by


title: Eigenvalues and eigenvectors

text: that is, acceleration is proportional to position (i.e., we expect formula_234 to be sinusoidal in time).


title: Eigenvalues and eigenvectors

text: In formula_57 dimensions, formula_236 becomes a mass matrix and formula_70 a stiffness matrix. Admissible solutions are then a linear combination of solutions to the generalized eigenvalue problem


title: Eigenvalues and eigenvectors

text: where formula_239 is the eigenvalue and formula_240 is the (imaginary) angular frequency. The principal vibration modes are different from the principal compliance modes, which are the eigenvectors of formula_70 alone. Furthermore, damped vibration, governed by


title: Eigenvalues and eigenvectors

text: This can be reduced to a generalized eigenvalue problem by algebraic manipulation at the cost of solving a larger system.


title: Eigenvalues and eigenvectors

text: The orthogonality properties of the eigenvectors allows decoupling of the differential equations so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using finite element analysis, but neatly generalize the solution to scalar-valued vibration problems.


title: Eigenvalues and eigenvectors

text: In image processing, processed images of faces can be seen as vectors whose components are the brightnesses of each pixel. The dimension of this vector space is the number of pixels. The eigenvectors of the covariance matrix associated with a large set of normalized pictures of faces are called eigenfaces; this is an example of principal component analysis. They are very useful for expressing any face image as a linear combination of some of them. In the facial recognition branch of biometrics, eigenfaces provide a means of applying data compression to faces for identification purposes. Research related to eigen vision systems determining hand gestures has also been made.


title: Eigenvalues and eigenvectors

text: Similar to this concept, eigenvoices represent the general direction of variability in human pronunciations of a particular utterance, such as a word in a language. Based on a linear combination of such eigenvoices, a new voice pronunciation of the word can be constructed. These concepts have been found useful in automatic speech recognition systems for speaker adaptation.


title: Eigenvalues and eigenvectors

text: In mechanics, the eigenvectors of the moment of inertia tensor define the principal axes of a rigid body. The tensor of moment of inertia is a key quantity required to determine the rotation of a rigid body around its center of mass.


title: Eigenvalues and eigenvectors

text: In solid mechanics, the stress tensor is symmetric and so can be decomposed into a diagonal tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no shear components; the components it does have are the principal components.


title: Eigenvalues and eigenvectors

text: In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix formula_30, or (increasingly) of the graph's Laplacian matrix due to its discrete Laplace operator, which is either formula_245 (sometimes called the "combinatorial Laplacian") or formula_246 (sometimes called the "normalized Laplacian"), where formula_33 is a diagonal matrix with formula_248 equal to the degree of vertex formula_249, and in formula_250, the formula_134th diagonal entry is formula_252. The formula_70th principal eigenvector of a graph is defined as either the eigenvector corresponding to the formula_70th largest or formula_70th smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector.


title: Eigenvalues and eigenvectors

text: The principal eigenvector is used to measure the centrality of its vertices. An example is Google's PageRank algorithm. The principal eigenvector of a modified adjacency matrix of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the stationary distribution of the Markov chain represented by the row-normalized adjacency matrix; however, the adjacency matrix must first be modified to ensure a stationary distribution exists. The second smallest eigenvector can be used to partition the graph into clusters, via spectral clustering. Other methods are also available for clustering.


title: Eigenvalues and eigenvectors

text: The basic reproduction number (formula_256) is a fundamental number in the study of how infectious diseases spread. If one infectious person is put into a population of completely susceptible people, then formula_256 is the average number of people that one typical infectious person will infect. The generation time of an infection is the time, formula_258, from one person becoming infected to the next person becoming infected. In a heterogeneous population, the next generation matrix defines how many people in the population will become infected after time formula_258 has passed. formula_256 is then the largest eigenvalue of the next generation matrix.


title: Quinoa

text: Quinoa ("Chenopodium quinoa"; , from Quechua ' or ') is a flowering plant in the amaranth family. It is a herbaceous annual plant grown as a crop primarily for its edible seeds; the seeds are rich in protein, dietary fiber, B vitamins, and dietary minerals in amounts greater than in many grains. Quinoa is not a grass, but rather a pseudocereal botanically related to spinach and amaranth ("Amaranthus" spp.), and originated in the Andean region of northwestern South America. It was first used to feed livestock 5,2007,000 years ago, and for human consumption 3,0004,000 years ago in the Lake Titicaca basin of Peru and Bolivia.


title: Quinoa

text: The plant thrives at high altitudes and produces seeds that are rich in protein. Almost all production in the Andean region is done by small farms and associations. Its cultivation has spread to more than 70 countries, including Kenya, India, the United States, and in European countries. As a result of increased popularity and consumption in North America, Europe, and Australasia, quinoa crop prices tripled between 2006 and 2014.


title: Quinoa

text: The species "Chenopodium quinoa" was first described by Carl Ludwig Willdenow (17651812), a German botanist who studied plants from South America, brought back by explorers Alexander von Humboldt and Aimé Bonpland.


title: Quinoa

text: The genus name "Chenopodium" is composed of two words coming from the Greek χήν,-νός, "goose" and πόδῖον, "podion" "little foot", or "goose foot", because of the resemblance of the leaves with the trace of a crow's feet.


title: Quinoa

text: The specific epithet "quinoa" is a borrowing from the Spanish "quinua" or "quinoa", itself derived from Quechua "kinwa".


title: Quinoa

text: "Chenopodium quinoa" is a dicotyledonous annual plant, usually about high. It has broad, generally powdery, hairy, lobed leaves, normally arranged alternately. The woody central stem is branched or unbranched depending on the variety and may be green, red or purple. The flowering panicles arise from the top of the plant or from leaf axils along the stem. Each panicle has a central axis from which a secondary axis emerges either with flowers (amaranthiform) or bearing a tertiary axis carrying the flowers (glomeruliform). These are small, incomplete, sessile flowers of the same colour as the sepals, and both pistillate and perfect forms occur. Pistillate flowers are generally located at the proximal end of the glomeruli and the perfect ones at the distal end of it. A perfect flower has five sepals, five anthers and a superior ovary, from which two to three stigmatic branches emerge.


title: Quinoa

text: The green hypogynous flowers have a simple perianth and are generally self-fertilizing, though cross-pollination occurs. In the natural environment, betalains serve to attract animals to generate a greater rate of pollination and ensure, or improve, seed dissemination. The fruits (seeds) are about in diameter and of various colors — from white to red or black, depending on the cultivar.


title: Quinoa

text: In regards to the "newly" developed salinity resistance of "C.quinoa", some studies have concluded that accumulation of organic osmolytes plays a dual role for the species. They provide osmotic adjustment, in addition to protection against oxidative stress of the photosynthetic structures in developing leaves. Studies also suggested that reduction in stomatal density in reaction to salinity levels represents an essential instrument of defence to optimize water use efficiency under the given conditions to which it may be exposed.


title: Quinoa

text: "Chenopodium quinoa" is believed to have been domesticated in the Peruvian Andes from wild or weed populations of the same species. There are non-cultivated quinoa plants ("Chenopodium quinoa" var. "melanospermum") that grow in the area it is cultivated; these may either be related to wild predecessors, or they could be descendants of cultivated plants.


title: Quinoa

text: Raw, uncooked quinoa is 13% water, 64% carbohydrates, 14% protein, and 6% fat. Nutritional evaluations indicate that a serving of raw quinoa seeds is a rich source (20% or higher of the Daily Value, DV) of protein, dietary fiber, several B vitamins, including 46%DV for folate, and the dietary minerals magnesium, phosphorus, and manganese (table).


title: Quinoa

text: After boiling, which is the typical preparation for eating the seeds, quinoa is 72% water, 21% carbohydrates, 4% protein, and 2% fat. In a serving, "cooked" quinoa provides of food energy and is a rich source of manganese and phosphorus (30% and 22%DV, respectively), and a moderate source (10–19%DV) of dietary fiber, folate, and the dietary minerals iron, zinc, and magnesium (table).


title: Quinoa

text: Quinoa is gluten-free. Because of the high concentration of protein, ease of use, versatility in preparation, and potential for increased yields in controlled environments, it has been selected as an experimental crop in NASA's Controlled Ecological Life Support System for long-duration human occupied space flights.


title: Quinoa

text: In their natural state, the seeds have a coating that contains bitter-tasting saponins, making them unpalatable. Most of the grain sold commercially has been processed to remove this coating. This bitterness has beneficial effects during cultivation, as it deters birds and therefore, the plant requires minimal protection. The genetic control of bitterness involves quantitative inheritance. Although lowering the saponin content through selective breeding to produce sweeter, more palatable varieties is complicated by ≈10% cross-pollination, it is a major goal of quinoa breeding programs, which may include genetic engineering.


title: Quinoa

text: The toxicity category rating of the saponins in quinoa treats them as mild eye and respiratory irritants and as a low gastrointestinal irritant. In South America, these saponins have many uses, including as a detergent for clothing and washing, and as a folk medicine antiseptic for skin injuries.


title: Quinoa

text: Additionally, the leaves and stems of all species of the genus "Chenopodium" and related genera of the family Amaranthaceae, including quinoa, contain high levels of oxalic acid.


title: Quinoa

text: The plant's growth is highly variable due to the number of different subspecies, varieties and landraces (domesticated plants or animals adapted to the environment in which they originated). However, it is generally undemanding and altitude-hardy; it is grown from coastal regions to over in the Andes near the equator, with most of the cultivars being grown between and . Depending on the variety, optimal growing conditions are in cool climates with temperatures that vary between during the night to near during the day. Some cultivars can withstand lower temperatures without damage. Light frosts normally do not affect the plants at any stage of development, except during flowering. Midsummer frosts during flowering, a frequent occurrence in the Andes, lead to sterilization of the pollen. Rainfall requirements are highly variable between the different cultivars, ranging from during the growing season. Growth is optimal with well-distributed rainfall during early growth and no rain during seed maturation and harvesting.


title: Quinoa

text: Quinoa has been cultivated in the United States, primarily in the high elevation San Luis Valley of Colorado where it was introduced in 1983. In this high-altitude desert valley, maximum summer temperatures rarely exceed and night temperatures are about . In the 2010s, experimental production was attempted in the Palouse region of Eastern Washington, and farmers in Western Washington began producing the crop. The Washington State University Skagit River Valley research facility near Mount Vernon grew thousands of its own experimental varieties. The Puget Sound region's climate is similar to that of coastal Chile where the crop has been grown for centuries. Due to the short growing season, North American cultivation requires short-maturity varieties, typically of Bolivian origin. Quinoa is planted in Idaho where a variety developed and bred specifically for the high-altitude Snake River Plain is the largest planted variety in North America.


title: Quinoa

text: Quinoa plants do best in sandy, well-drained soils with a low nutrient content, moderate salinity, and a soil pH of 6 to8.5. The seedbed must be well prepared and drained to avoid waterlogging.


title: Quinoa

text: Quinoa has gained attention for its adaptability to contrasting environments such as saline soils, nutrient-poor soils and drought stressed marginal agroecosystems. Yields are maximised when of nitrogen is available. The addition of phosphorus does not improve yield.


title: Quinoa

text: In eastern North America, it is susceptible to a leaf miner that may reduce crop success. (The miner also affects the common weed and close relative "Chenopodium album", but "C.album" is much more resistant.)


title: Quinoa

text: Rotation is used in its Andean native range. Rotation is common with potato, cereals and legumes including "Lupinus mutabilis".


title: Quinoa

text: The genome of quinoa was sequenced in 2017. Through traditional selective breeding and, potentially, genetic engineering, the plant is being modified to have higher crop yield, improved tolerance to heat and biotic stress, and greater sweetness through saponin inhibition.


title: Quinoa

text: Traditionally, quinoa grain is harvested by hand, and only rarely by machine, because the extreme variability of the maturity period of most quinoa cultivars complicates mechanization. Harvest needs to be precisely timed to avoid high seed losses from shattering, and different panicles on the same plant mature at different times. The crop yield in the Andean region (often around 3 t/ha up to 5 t/ha) is comparable to wheat yields. In the United States, varieties have been selected for uniformity of maturity and are mechanically harvested using conventional small grain combines.


title: Quinoa

text: The plants are allowed to stand until the stalks and seeds have dried out and the grain has reached a moisture content below 10%.


title: Quinoa

text: Handling involves threshing the seedheads from the chaff and winnowing the seed to remove the husk. Before storage, the seeds need to be dried in order to avoid germination. Dry seeds can be stored raw, until being washed or mechanically processed to remove the pericarp to eliminate the bitter layer containing saponins. This was traditionally done manually, which is labour-intensive. The seeds must be dried again before being stored and sold in stores.


title: Quinoa

text: In 2020, world production of quinoa was 175,188 tonnes, led by Peru and Bolivia with 97% of the total when combined (table).


title: Quinoa

text: Since the early 21st century when quinoa became more commonly consumed in North America, Europe, and Australasia where it was not typically grown, the crop value increased. Between 2006 and 2013, quinoa crop prices tripled. In 2011, the average price was US $3,115 per tonne with some varieties selling as high as $8,000 per tonne. This compares with wheat prices of about US $340 per tonne, making wheat about 10% of the value of quinoa. The resulting effect on traditional production regions in Peru and Bolivia also influenced new commercial quinoa production elsewhere in the world, such as the United States. By 2013, quinoa was being cultivated in some 70 countries. As a result of expanding production outside the Andean highlands native for quinoa, the price plummeted starting in early 2015 and remained low for years. From 2018 to 2019, quinoa production in Peru declined by 22%. Some refer to this as the "quinoa bust" because of the devastation the price fall caused for farmers and industry.


title: Quinoa

text: Rising quinoa prices over the period of 2006 to 2017 may have reduced the affordability of quinoa to traditional consumers. However, a 2016 study using Peru's Encuesta Nacional de Hogares found that rising quinoa prices during 2004–2013 led to net economic benefits for producers, and other commentary indicated similar conclusions, including for women specifically. Impacts of the price surge on quinoa consumption in the Andes mainly affected urban poor rather than farmers themselves, and these impacts were reduced when the price fell in 2015. It has also been suggested that as quinoa producers rise above subsistence-level income, they switch their own consumption to Western processed foods which are often less healthy than a traditional, quinoa-based diet, whether because quinoa is held to be worth too much to keep for oneself and one's family, or because processed foods have higher status despite their poorer nutritional value. Efforts are being made in some areas to distribute quinoa more widely and ensure that farming and poorer populations have access to it and have an understanding of its nutritional importance, including use in free school breakfasts and government provisions distributed to pregnant and nursing women in need.


title: Quinoa

text: In terms of wider social consequences, research on traditional producers in Bolivia has emphasised a complex picture. The degree to which individual producers benefit from the global quinoa boom depends on its mode of production, for example through producer associations and co-operatives such as the Asociación Nacional de Productores de Quinua (founded in the 1970s), contracting through vertically-integrated private firms, or wage labor. State regulation and enforcement may promote a shift to cash-cropping among some farmers and a shift toward subsistence production among others, while enabling many urban refugees to return to working the land, outcomes with complex and varied social effects.


title: Quinoa

text: The growth of quinoa consumption outside of its indigenous region has raised concerns over food security of the original consumers, unsustainably intensive farming of the crop, expansion of farming into otherwise marginal agricultural lands with concurrent loss of the natural environment, threatening both the sustainability of producer agriculture and the biodiversity of quinoa.


title: Quinoa

text: World demand for quinoa is sometimes presented in the media particularly as being caused by rising veganism, but one academic has commented that despite the drawbacks of quinoa, meat production in most cases is still less sustainable than quinoa.


title: Quinoa

text: The United Nations General Assembly declared 2013 as the "",<ref name="UN Resolution 66/221. International Year of Quinoa, 2013"></ref> in recognition of the ancestral practices of the Andean people, who have preserved it as a food for present and future generations, through knowledge and practices of living in harmony with nature. The objective was to draw the world’s attention to the role that quinoa could play in providing food security, nutrition and poverty eradication in support of achieving Millennium Development Goals. Some academic commentary emphasized that quinoa production could have ecological and social drawbacks in its native regions, and that these problems needed to be tackled.


title: Quinoa

text: Quinoa is used in the Jewish community as a substitute for the leavened grains that are forbidden during the Passover holiday. Several kosher certification organizations refuse to certify it as being kosher for Passover, citing reasons including its resemblance to prohibited grains or fear of cross-contamination of the product from nearby fields of prohibited grain or during packaging. However, in December 2013 the Orthodox Union, the world's largest kosher certification agency, announced it would begin certifying quinoa as kosher for Passover.


title: Quinoa

text: Quinoa is an allotetraploid plant, containing two full sets of chromosomes from two different species which hybridised with each other at one time. According to a 1979 study, its presumed ancestor is either "Chenopodium berlandieri", from North America, or the Andean species "Ch. hircinum", although more recent studies, in 2011, even suggest Old World relatives. On the other hand, morphological features relate "Ch.quinoa" of the Andes and "Ch. nuttalliae" of Mexico. Some studies have suggested that both species may have been derived from the same wild type. A weedy quinoa, "Ch.quinoa" var. "melanospermum", is known from South America, but no equivalent closely related to "Ch.nutalliae" has been reported from Mexico so far.


title: Quinoa

text: Studies regarding the genetic diversity of quinoa suggest that it may have passed through at least three bottleneck genetic events, with a possible fourth expected:


title: Quinoa

text: Over the last 5,000years the biogeography of "Ch. quinoa" has changed greatly, mainly by human influence, convenience and preference. It has changed not only in the area of distribution, but also in regards to the climate this plant was originally adapted to, in contrast to the climates on which it is able to do successfully grow in now. In a process started by a number of pre-Inca South American indigenous cultures, people in Chile have been adapting quinoa to salinity and other forms of stress over the last 3,000years. Quinoa is also cultivated, since an early date, near the coast of northern Chile, where was grown by the Chinchorro culture. "Ch. quinoa" was brought to the lowlands of south-central Chile at an early date from the Andean highlands. Varieties in the lowlands of south-central Chile derive directly from ancestral cultivars which then evolved in parallel to those of the highlands. It has been suggested that the introduction of "Ch. quinoa" occurred before highland varieties with floury perisperm emerged. There are wide discrepancies in the suggested dates of introduction, one study suggest c. 1000 BC as introduction date while another 600–1100 AD. In colonial times the plant is known to have been cultivated as far south as Chiloé Archipelago and the shores of Nahuel Huapi Lake. The cuisine of Chiloé included bread made of Quinoa until at least the mid-19th century.


title: Quinoa

text: In Chile it had almost disappeared by the early 1940s; as of 2015 the crop is mostly grown in three areas by only some 300 smallholder farmers. Each of these areas is different: indigenous small-scale growers near the border with Bolivia who grow many types of Bolivian forms using the Inca "ayllu" clan system, a few farmers in the central region who exclusively grow a white-seeded variety and generally market their crops through a well-known cooperative, and in the south by women in home gardens in Mapuche reserves.


title: Quinoa

text: When Amaranthaceae became abundant in Lake Pacucha, Peru, the lake was fresh, and the lack of Amaranthaceae taxa strongly indicates droughts which turned the lake into a saltmarsh. Based on the pollen associated with soil manipulation, this is an area of the Andes where domestication of "C.quinoa" became popular, although it was not the only one. It was domesticated in various geographical zones. With this, morphological adaptations began to happen until having five ecotypes today. Quinoa's genetic diversity illustrates that it was and is a vital crop.


title: Quinoa

text: Andean agronomists and nutrition scientists began researching quinoa in the early twentieth century, and it became the subject of much interest among researchers involved in neglected and underutilized crop studies in the 1970s. The grain, however, has received much less attention than crops like maize or wheat.


title: Quinoa

text: Particularly for the high variety of Chilean landraces, in addition to how the plant has adapted to different latitudes, this crop is now potentially cultivable almost anywhere in the world.


title: Snatch (film)

text: Snatch (stylized as snatch.) is a 2000 British-American crime comedy film written and directed by Guy Ritchie, featuring an ensemble cast. Set in the London criminal underworld, the film contains two intertwined plots: one dealing with the search for a stolen diamond, the other with a small-time boxing promoter (Jason Statham) who finds himself under the thumb of a ruthless gangster (Alan Ford) who is ready and willing to have his subordinates carry out severe and sadistic acts of violence.


title: Snatch (film)

text: The film features an assortment of characters, including Irish Traveller "One Punch" Mickey O'Neil (Brad Pitt), referred to as a "Pikey", Russian arms-dealer Boris "the Blade" Yurinov (Rade Šerbedžija), professional thief and gambling addict Franky "Four-Fingers" (Benicio del Toro), American gangster-jeweller Abraham Denovitz known as "Cousin Avi" (Dennis Farina), small-time crooks Sol (Lennie James) and Vinny (Robbie Gee), getaway driver Tyrone (Ade), and bounty hunter Bullet-Tooth Tony (Vinnie Jones). It is also distinguished by a kinetic direction and editing style, an intricate double plot featuring numerous ironic twists of chance and causality, and a fast pace.


title: Snatch (film)

text: The film shares themes, ideas, and motifs with Ritchie's first film, "Lock, Stock and Two Smoking Barrels". It is also filmed in the same visual style and features many of the same actors, including Vinnie Jones, Jason Statham, Jason Flemyng, and Alan Ford.


title: Snatch (film)

text: After stealing an diamond while dressed as an ultra-Orthodox Jew during a heist in Antwerp, Franky "Four-Fingers" goes to London to see diamond dealer Doug "The Head" on behalf of New York jeweler and Jewish-American organized crime figure "Cousin Avi". One of the other robbers advises Franky to obtain a gun from his brother, arms dealer and ex-KGB agent Boris "The Blade", then later calls Boris and encourages him to steal the diamond from Franky before he can turn it over to Doug.


title: Snatch (film)

text: Meanwhile, Cockney boxing promoter and slot machine shop owner Turkish is persuaded by crime boss "Brick Top" to put his boxer "Gorgeous George" in a match against one of Brick Top's boxers. However, when Turkish sends his partner Tommy and Gorgeous George to purchase a caravan from a clan of Irish Travellers, George gets challenged to a fistfight against Traveller bare-knuckle boxing champion Mickey O'Neil, who beats up and severely injures George. Turkish persuades Mickey to replace George in his upcoming match by agreeing to purchase a new caravan for Mickey's mother. Brick Top grudgingly agrees, but only on the condition that Mickey will throw the fight in the fourth round.


title: Snatch (film)

text: Boris gives Franky a revolver in exchange for a favour: Franky is to place a bet on Boris' behalf at Brick Top's bookies. Avi, knowing Franky has gambling addiction, flies to London with his bodyguard "Rosebud" to pick up the diamond personally. Boris hires Vinny and Sol, two pawnbrokers and small-time crooks, to rob Franky while he is at the bookies. The robbery goes awry when they crash their car into Franky's van while trying to park, trapping Franky inside. Sol, Vinny, and their getaway driver Tyrone are caught on camera and find no money at the bookies due to the bets being cancelled since Gorgeous George had to drop out, but manage to kidnap Franky in their escape. At their pawn shop, Sol and Vinny hold Franky captive with a sack over his head. Upon Boris' arrival, Sol and Vinny demand he gives them half of the cash when he sells the diamond, during which Vinny utters Boris' name. Realizing that his identity and betrayal has been exposed to Franky, Boris kills Franky by shooting him in the head, and leaves with the diamond.


title: Snatch (film)

text: Instead of throwing the fight, Mickey accidentally knocks his opponent out with a single punch due to his overwhelming power. Infuriated, Brick Top robs Turkish of his life savings and demands that Mickey fight again, and lose since the majority of the gamblers will now bet on him. Mickey refuses to fight again unless Turkish buys an even better caravan for his mother, but Turkish has no money left since Brick Top stole his savings. Furious, Brick Top has his men vandalize Turkish's gambling arcade and burn down Mickey's mother's caravan while she is asleep inside. Brick Top and his men then tracks down Tyrone, Sol and Vinny to kill them for robbing his bookies. Sol bargains for their lives by offering Brick Top the stolen diamond, and is given 48 hours to retrieve it.


title: Snatch (film)

text: Avi and Doug hire "Bullet-Tooth" Tony to help them find Franky. When the trail leads to Boris, they kidnap him and retrieve the diamond, while being closely pursued by Sol, Vinny, and Tyrone. Turkish and Tommy, who are on their way to purchase a gun from Boris, are driving on the same stretch of road at the time. When Tommy throws Turkish's carton of milk out of their car window, it splashes over Tony's windscreen, causing him to crash which accidentally kills Rosebud in the process. Boris escapes from the wreck only to be hit by Tyrone's car. Tony and Avi are confronted by Sol, Vinny, and Tyrone at a pub where Tony realizes that the trio's pistols are replicas, which he contrasts with his real handgun, intimidating them into leaving. The wounded Boris arrives with an assault rifle and a grenade launcher looking for the diamond, but is shot and killed by Tony, who wounds Tyrone at the same time. Sol and Vinny leave a wounded Tyrone and escape with the diamond, which Vinny hides in his pants. When Tony catches up to them, they tell him that the diamond is back at their pawn shop. Once there, Vinny pretends to have misplaced the diamond, then accuses his dog, which he got earlier from the Irish Traveller clan, of eating it. When Avi tells Tony to kill the dog, Vinny gives in and produces the diamond from his pants, but the dog snatches the diamond away and runs off, presumably back to the Irish Travellers' campsite. Avi wildly fires at the fleeing dog, accidentally killing Tony. He gives up and returns to New York City.


title: Snatch (film)

text: Mickey agrees to fight to avoid more carnage, but gets so drunk after his mother's wake that Turkish fears he will not make it to the fourth round. If he fails to go down in the fourth round as agreed, Brick Top vows that his men will murder Turkish, Tommy, Mickey, and his entire clan of Travellers.