matrix mathematics mathematics matrix rectangular array numbers symbols expressions arranged rows columns dimensions matrix read rows columns individual items matrix called elements entries provided size number rows number columns added element element rule matrix multiplied number columns equals number rows second matrix multiplied element wise associated field major application represent linear linear functions example rotation dimensional space linear transformation represented rotation matrix column matrix column describing position point space product column describing position point rotation product transformation matrix represents composition linear application solution systems linear matrix square possible properties computing example square matrix zero insight geometry linear transformation information matrix applications found scientific fields branch physics including classical mechanics quantum mechanics quantum study physical phenomena motion rigid bodies computer graphics project model dimensional screen probability theory statistics describe sets instance ranks pages google search matrix classical notions derivatives higher dimensions major branch numerical analysis devoted development efficient matrix subject centuries old today expanding area research matrix methods theoretically practically tailored particular matrix structures sparse near diagonal finite element method infinite occur theory atomic theory simple example infinite matrix matrix representing derivative operator acts taylor series function definition matrix rectangular array numbers mathematical objects operations addition defined commonly matrix field rectangular array member article focuses real complex elements real numbers complex numbers respectively general types entries discussed instance real matrix formula numbers symbols expressions matrix called entries elements horizontal vertical lines entries matrix called rows columns respectively size size matrix defined number rows columns contains matrix rows columns called matrix matrix called dimensions example matrix matrix single row called row single column called column matrix number rows columns called square matrix matrix infinite number rows columns called infinite matrix contexts computer programs useful consider matrix rows columns called matrix commonly written box formula symbolic matrix vary widely prevailing trends usually upper case letters examples corresponding lower case letters represent entries addition upper case letters symbolize authors use special style commonly upright non distinguish mathematical objects alternative involves use double variable style formula entry row column matrix referred entry matrix commonly denoted alternative entry example entry following matrix denoted formula entries matrix defined formula example entries following matrix determined formula case matrix defined formula square double example matrix defined matrix size mentioned formula valid specified separately instance matrix defined programming languages utilize represent matrix programming languages start numbering array zero case entries matrix article follows common convention mathematical writing starts set denoted basic operations number basic operations applied modify called matrix addition matrix row operations addition familiar properties numbers extend operations example addition matrix sum depend order compatible addition expressed finally matrix defined number columns left matrix number rows right matrix matrix matrix matrix product matrix entries given dot product corresponding row corresponding column formula example entry product calculated formula matrix rules left right size products defined product defined defined respectively products defined need equal generally matrix marked contrast rational real complex numbers product independent order factors example formula formula ordinary matrix described exist frequently operations considered forms product product arise solving matrix equation row operations types row operations row addition adding row row entries row non zero constant row switching rows matrix operations number ways including solving linear finding matrix matrix obtained collection rows columns example following matrix construct removing row column formula minors matrix found computing certain principal square obtained removing certain rows columns definition varies author author according authors principal set row remain set column remain authors define principal rows columns number ones remain type called leading principal linear write work multiple linear systems linear example matrix column matrix column matrix equation equivalent system linear linear matrix reveal essential features related linear known linear maps real matrix gives rise linear transformation mapping matrix product conversely linear transformation arises unique matrix explicitly coordinate unit position matrix said represent linear map called transformation matrix example matrix formula viewed transform unit square pictured right obtained column formula formula turn define unit square following table shows number associated linear maps blue original mapped green grid shapes origin marked black point correspondence linear maps matrix corresponds composition maps matrix represents linear map composition represented equality follows mentioned matrix rank matrix maximum number independent row matrix maximum number independent column dimension image linear map represented rank states dimension matrix plus rank equals number columns matrix square matrix square matrix matrix number rows columns matrix known square matrix order square order added multiplied entries form main diagonal square matrix lie imaginary line runs left corner right corner matrix main types diagonal triangular matrix entries main diagonal zero called upper triangular matrix similarly entries main diagonal zero called lower triangular matrix diagonal elements zero called diagonal matrix identity matrix identity matrix size matrix elements main diagonal equal elements equal formula square matrix order special kind diagonal matrix called identity matrix leaves matrix unchanged ain matrix matrix square matrix equal matrix instead equal negative matrix complex symmetry replaced concept satisfy star denotes matrix complex real complex linear combination cases real infinite dimensional situations related infinitely rows columns matrix square matrix called non singular exists matrix exists unique called matrix denoted definite matrix matrix called positive definite respectively negative definite indefinite associated form given takes positive values respectively negative values negative positive values form takes non negative respectively non positive values matrix called positive respectively negative matrix indefinite precisely positive negative matrix positive definite positive matrix positive table right shows possibilities allowing input different instead yields form associated matrix matrix square matrix real entries columns rows unit matrix equal formula formula identity matrix size matrix necessarily normal matrix special matrix matrix linear transformation matrix pure rotation matrix pure reflection composition reflection rotation complex matrix matrix main operations trace trace square matrix sum diagonal entries matrix mentioned trace product independent order factors immediate definition matrix formula trace matrix equal square matrix number certain properties matrix matrix absolute value equals area volume image unit square cube sign corresponds orientation corresponding linear map positive orientation preserved given formula involves terms rule lengthy formula dimensions product square equals product adding multiple row row multiple column column change rows columns affects operations matrix transformed lower upper triangular matrix equals product entries main diagonal provides method matrix finally expansion expresses terms minors smaller expansion definition taking starting case matrix unique entry matrix seen equivalent formula solve linear systems rule division related square value system number non zero satisfying called respectively number matrix equivalent formula given evaluation called characteristic degree equation different solutions matrix complex entries real according hamilton result matrix characteristic yields zero matrix aspects matrix calculations performed different techniques problems solved direct approaches example square matrix obtained finding sequence tends infinity able choose appropriate specific problem important determine effectiveness precision available domain studying matters called numerical linear numerical situations main aspects complexity numerical stability determining complexity means finding upper bounds estimates elementary operations additions necessary perform example calculating matrix product matrix definition given needs entries product necessary naive needs refined approach incorporates specific features computing devices practical situations additional information involved known important case sparse entries zero specifically adapted solving linear systems sparse gradient method roughly speaking numerically stable little input values lead big result example calculating matrix formula denotes matrix lead significant rounding errors matrix small norm matrix capture conditioning linear problems computing matrix computer languages designed commands libraries early 1970s engineering computers rom add basic commands computer languages designed manipulate mathematical programs aid computing methods render easily accessible form generally referred matrix matrix techniques interest techniques preserve certain properties question rank quantities calculated applying transformation certain matrix operations easier carry types factors product lower upper triangular calculated linear systems solved efficiently simple technique called forward likewise triangular easier elimination similar transforms matrix row form methods proceed matrix suitable elementary correspond rows columns adding row row singular value expresses matrix product diagonal matrix expresses product diagonal matrix suitable matrix written form called generally applicable jordan transforms matrix jordan normal form entries placed main diagonal possibly entries equal directly main diagonal shown right given power fold matrix calculated power diagonal matrix calculated taking corresponding powers diagonal entries easier instead matrix need frequently arising solving linear differential matrix square roots avoid numerically ill situations employed abstract aspects different ways abstract uses entries general fields rings linear properties notion linear maps possible consider infinitely columns rows extension seen higher dimensional numbers opposed realised sequences numbers rectangular dimensional numbers subject certain requirements tend form groups known matrix groups similarly certain conditions form rings known matrix rings product general certain form fields known matrix fields general entries article focuses entries real complex numbers considered general types entries real complex numbers step field set addition division operations defined behaved instead example rational numbers finite fields example theory makes use finite fields considered roots exist larger field entries matrix instance complex case matrix real entries possibility entries matrix elements larger field view real matrix complex matrix entries happen real allows considering square matrix possess set alternatively consider entries closed field outset generally abstract makes great use entries ring general notion fields division operation need exist addition operations extend setting set square ring called matrix ring ring left module ring square ring defined formula matrix situation field element called entries ring ring special common case block considered entries entries need need members ordinary ring sizes certain compatibility conditions relationship linear maps linear maps equivalent described generally linear map finite dimensional spaces described matrix choosing bases dimension dimension formula words column expresses image terms basis relation uniquely determines entries matrix matrix depends choice bases different choices bases rise different equivalent concrete notions light example matrix describes linear map given respect dual bases properties natural way category entries field formula composition equivalent category finite dimensional spaces linear maps field generally set represent linear maps free arbitrary ring unity composition maps possible gives rise matrix ring representing ring matrix groups group mathematical structure consisting set objects operation operation combining objects subject certain requirements group objects group operation matrix called matrix group group element general matrix groups groups given size called general linear groups property preserved matrix products define matrix groups example given size form smaller group contained general linear group called special linear group determined condition form group matrix form called special group finite group matrix group considering regular representation group general groups studied matrix groups comparatively understood means representation theory infinite possible consider infinitely rows columns infinite objects write explicitly matters element set rows element set columns defined entry index sets need natural numbers basic operations addition defined problem matrix involve infinite define resulting entries defined general ring unity ring formula right module ring column finite formula entries formula columns contain entries considered left module result analogous object row finite formula rows entries infinite describe linear maps columns finite number entries following reason matrix describe linear map bases spaces chosen recall definition means space written uniquely finite linear combination basis written column entries columns describe images individual basis basis meaningful columns entries restriction rows product involved entries given infinite sum products involves terms defined amounts forming linear combination columns effectively involves result entries columns sees products given type defined provided usual column index row index sets match type corresponds composition linear maps ring condition row column relaxed norm place absolutely series instead finite sums example column sums absolutely sequences form ring course row sums absolutely series form ring vein infinite describe operators spaces convergence continuity questions arise results certain constraints imposed explicit point view tends matter abstract powerful tools functional analysis instead matrix matrix number rows columns zero help dealing maps involving zero space example matrix matrix zero matrix corresponding map dimensional space matrix common computer systems allow creating computing matrix follows product occurring formula value consistent fact identity map finite dimensional space fact characterization applications numerous applications mathematics sciences merely advantage compact representation set numbers matrix example game theory economics matrix players depending given finite set alternatives players choose text mining automated compilation makes use document term track frequencies certain words documents complex numbers represented particular real formula addition complex numbers correspond example rotation represent complex number absolute value similar interpretation possible general early techniques hill linear nature codes comparatively easy break computer graphics uses represent objects objects rotation accomplish tasks projecting dimensional object dimensional screen corresponding theoretical camera observation ring important study control theory chemistry makes use ways particularly use quantum theory discuss molecular bonding examples overlap matrix matrix solving obtain molecular method theory matrix finite basic notion theory records connected edge containing different values meaning example yes respectively called logical distance cost matrix contains information distances edges concepts applied websites connected cities connected roads etc case connection network extremely dense tend sparse contain entries specifically tailored matrix network theory analysis geometry matrix function consists second derivatives respect coordinate directions formula information local growth behaviour function given critical point point partial derivatives formula function local minimum matrix positive definite programming find global functions closely related ones attached matrix frequently situations matrix map denote components matrix defined formula rank matrix value locally point implicit function partial differential classified considering matrix highest order differential operators equation partial differential matrix positive definite decisive influence set possible solutions equation question finite element method important numerical method solve partial differential widely applied complex physical systems attempts approximate solution equation linear functions pieces chosen respect sufficiently fine grid turn recast matrix equation probability theory statistics square rows probability entries non negative sum define chains states row matrix gives probability distribution position particle currently state corresponds row properties chain like absorbing states states particle eventually read transition statistics makes use different forms descriptive statistics concerned describing data sets represented data subjected reduction techniques matrix mutual random technique linear squares method finite set pairs linear function formulated terms related singular value random entries random numbers subject suitable probability matrix normal distribution probability theory applied domains ranging number theory physics physics linear associated play key role modern physics example elementary particles quantum field theory classified representations group special specifically behavior spin group concrete representations involving general gamma integral physical description behave group theoretical representation involving special group calculations use convenient matrix representation known gauge group forms basis modern description strong nuclear interactions quantum matrix turn expresses fact basic states important weak interactions related basic states define particles specific distinct masses linear combinations quantum states model quantum mechanics represented theory operators infinite dimensional acting quantum states referred matrix mechanics particular example density matrix characterizes mixed state quantum system linear combination elementary pure matrix serves key tool describing scattering experiments form cornerstone experimental particle physics collision reactions occur particle non interacting particles head small interaction zone new set non interacting particles result described product outgoing particle states linear combination particle states linear combination given matrix known matrix information possible interactions particles normal modes general application physics description coupled systems motion systems described matrix form mass matrix velocity term force matrix displacement characterize interactions best way obtain solutions determine system normal modes matrix equation techniques like crucial comes internal dynamics molecules internal vibrations systems consisting mutually bound component needed describing mechanical vibrations electrical circuits provides matrix applications theory wave nature light neglected result model light rays rays light rays optical elements small action lens reflective element given light ray expressed component matrix called ray transfer matrix components light ray slope distance optical axis matrix properties optical element actually kinds matrix describing lens surface translation matrix describing translation plane reference surface matrix applies optical system consisting combination lenses reflective elements simply described matrix resulting product components electronics traditional analysis analysis electronics lead system linear described matrix behaviour electronic components described let dimensional component input voltage input current elements let dimensional component output voltage output current elements behaviour electronic component described matrix containing element element elements calculating circuit reduces history long history application solving linear known 1800s chinese text chapters mathematical art written 10th 2nd century bce example use array methods solve simultaneous including concept italian mathematician brought method europe published japanese mathematician array methods solve simultaneous represented book elements curves publicized use recording information solutions experimented different systems presented rule term matrix latin derived mater mother coined understood matrix object giving rise number today called minors smaller derive original removing columns rows paper explains previous papers defined matrix rectangular array terms different systems common parent published treatise geometric rotated versions investigated previously instead defined operations addition division showed properties held true investigated demonstrated non property matrix property matrix addition early matrix theory limited use exclusively abstract matrix operations revolutionary instrumental proposing matrix concept independent equation systems published memoir theory proposed demonstrated english mathematician named use modern simultaneously demonstrated significant use represent matrix refers row column study sprang sources number theoretical problems led relate forms expressions linear maps dimensions developed notions including remark modern matrix products non prove general statements definition matrix following replace powers formula denotes product indicated terms showed real studied functional later called describe geometric local level die published treated opposed previous concrete approaches mentioned formula point firmly established established small example hamilton proved aforementioned memoir hamilton working forms dimensions end 19th century jordan elimination special case known elimination established jordan early 20th century attained central role linear partially use classification number systems previous century inception matrix mechanics born jordan led studying infinitely rows columns later von carried mathematical quantum mechanics developing functional notions linear operators spaces roughly speaking correspond space infinity independent directions historical word matrix mathematics word unusual ways authors historical importance use word matrix context proposed means reduce function lower type order function identical extension let matrix function involve apparent possible function matrix derived matrix means considering proposition asserts function question true possible values value arguments argument arguments remaining example function reduced collection functions single variable considering function possible values individuals substituted place variable resulting collection functions single variable reduced matrix values considering function possible values individuals substituted place variable introduction logic word matrix notion truth table mathematical logic diagonal element element differ zero diagonal matrix complex matrix matrix containing complex numbers sparse matrix sparse matrix physics references foundations applications isbn cliff moore guy standard model isbn isbn hill isbn riley mathematical methods physics engineering isbn 3rd volume foundations isbn isbn paul applications random physics nato science series mathematics physics chemistry berlin new_york isbn historical references memoir theory trans math papers introduction higher new_york dover publications isbn reprint original edition arthur collected mathematical papers jean des paris thomas theory theory historical intersection history mathematics boston berlin 1st berlin new_york isbn chapters companion commentary 2nd isbn collected works
Matrix (mathematics)