Now the standard method of finding eigenvalues and eigenvectors relies on finding determinants. And as a result, it's painful. So can we find eigenvalues and eigenvectors without determinants? The answer is, well I hope so, otherwise this will be a very short video. The path to finding eigenvectors without determinants begins with the following. So suppose v is any non-zero vector. Let's show that if vav is a dependent set, then v is an eigenvector. And while we're at it, let's find the corresponding eigenvalue. So remember, definitions are the whole of mathematics, all else is commentary, and this problem talks about dependent sets and eigenvectors, so it'll help to have the definition of dependent set and eigenvector. So let's pull those in. So remember, a set of vectors is said to be dependent if one or more of the vectors in our set can be expressed as a linear combination of the others. Since there's only two vectors in our set, then we can write one of them in terms of the other one. So suppose our vector v is c times the other vector av. If c is equal to zero, our vector v is also equal to zero, but that's not true. We've been assuming that v is a non-zero vector. So c can't be equal to zero, and so we can divide whatever c times v is equal to av. And if it's not written down, it didn't happen. Let's go ahead and write that. If vav is dependent, then something times v must be a times v, whereas something is usually some real number. Again, definitions are the whole of mathematics. All else is commentary. This problem also asks about eigenvectors, so let's pull in that definition. And we see that this is actually our definition of what makes up an eigenvector, eigenvalue pair. So v is an eigenvector with eigenvalue a, whatever that scalar multiple is. Well, suppose we have a independent set vav, but if we add in the vector a squared v, we get a dependent set. And let's show that there have to be scalars r1 and r2, where a minus r1i times a minus r2i is equal to the zero vector. Now, this is quite a handful, so let's see if we can make this a little bit easier to understand.  So we'll expand that. Now, because we're dealing with matrices, there are certain things we can do and certain things we can't do. So here we have to have this product ai, which we get from multiplying this a by this r2i. And we also have this product r1i a, which we get from multiplying this r1i by this a. But because matrix multiplication is not in general commutative, we can't equate ai with ia. Or can we? Remember, i is the identity matrix, the do nothing matrix. So this is do nothing and then do a. Well, this is do a, then do nothing. Well, in both cases, you're just doing a, so ai is a, and ia is a. But wait, there's more. This is i squared. Well, that's i times i, that's do nothing. And then do nothing. Well, that's really just the same as doing nothing, so this i squared is just i. We still have the a squared. And I'm subtracting r2 minus r1a, and so I can rewrite that. And since r1 and r2 are just elements of our field, they are other elements. This minus r1 plus r2, well that's really something we'll call that p. And this r1r2, that's something else we'll just call that q. And what that means is if I have an expression like this, I can factor the left-hand side as an expression like this. All right, let's put it together. Since v, av, a squared v are dependent, we must be able to express one of these in terms of the others. Now, if vav was an independent set, then the new vector, a squared v, must be the one we can express in terms of the others.  Now that can rearrange that a little bit. And since everything is right multiplied by v, I can factor that. But now I have this expression, and given some expression like a squared plus pa plus qi, I can factor it as a minus r1i, a minus r2i for some scalars r1 and r2. I can factor our matrix expression, which is what we wanted to show. So now suppose v is any non-zero vector, and we have this matrix equation, let's show that either v is an eigenvector for lambda2 or a minus lambda2 if is an eigenvector for lambda1. So definitions are the whole of mathematics. All else is commentary. Let's bring in our definition for an eigenvector eigenvalue. The useful thing we can do about mathematics is to engage in wishful thinking. This is sometimes known as a proof by cases. In this particular case, our wishful thinking might be this. Either a minus lambda2iv is the zero vector, or it isn't. So suppose a minus lambda2iv is the zero vector. We can expand that out. Now remember i is the do-nothing transformation. So this iv is really the same thing as v. And we can rearrange. And this is exactly our definition of what an eigenvector eigenvalue is. And so v is an eigenvector for lambda2. What this comes down to is this. Sometimes we're fantastically lucky, and the vector we pick happens to be the eigenvector. And if you want, you could live your entire life based on luck. Maybe you got lucky and got born to multimillionaire parents and went on to become the most powerful con man in history. But if you don't want to rely on luck, you have to rely on planning. And planning centers around considering all possibilities. So here, either this product is zero, or it isn't. So suppose a minus lambda2iv is w, where w is not the zero vector. Well, we still have our equation, and that gives me this equation, which we can expand. Again, i is the do-nothing transformation, so iw is the same as w. We don't need to include this i. And rearranging gives us the eigenvector eigenvalue relationship again. And so w, this a minus lambda2iv, is an eigenvector for lambda1.


hit record I just did this hit record monkey dropped me I think the other guy wasn't feeling well so he messaged text me last night and said that he wasn't going to set up the day so I just came down there in the street monkeys got a short day it's not supposed to rain it's nice outside here see nice day so it shouldn't rain so monkey went got me some change and then she went back at the house got her cigarette cigarette she don't smoke she went up got her coffee and she just went past a minute ago oh five minutes ago now but here's some stuff we've got we've got these things I saw one of these online the other day at these prices I come down on because I like to wiggle with people you know I'll come down on all these prices most of like these gloves they're only buck a piece these are two four dollars these are cool there's some visor things put your money or whatever your wallet you can put your oh what do you call it a registration in there if you want I brought these down I didn't bring the blower that I fixed yesterday and I don't know why I didn't but we'll get this airsoft gun here and all these BBs so if you buy the gun you get one of these what that these are five these are two now sell the whole crate for 30 bucks or 63 dollars worth of stuff in these are two dollars I got four of them two dollars a piece or all four for five that those got a couple clocks those are cool key chains yeah you can it holds a adjust to hold two to 12 keys it's also got a light got these knives I've got 80 knives here those are pretty cool they've got a sheath a little carry case with them these ones are nice here and of course you got your cheap ones I've got I got another one of these looks like a shotgun shell looks like a 12-gauge shotgun shell but it's a pocket knife these are just cheap ones here well I got a buck two dollars on that one this one's pretty cool it's got the flames on it so yeah they're they're pretty decent these are all dollar a piece so a couple movies here dollar a piece another camera there's a camera I saved one of those I got to the other day for those of you guys that didn't see them the video I gave this guy a dollar for a couple of things he had a dollar a piece on stuff he said damn can't get the light right he said uh bringing stuff over bringing stuff over here take this take that I still got a bunch of stuff up the house he gave me about 70 or 80 dollars worth of stuff for $1 that's pretty cool and already sold one of knives I was saying in my last video a bunch of these I had some of these but there was like I think I may have said 37 knives that anyway I've got 80 knives in total I got these they're five bucks a piece I got all these a bunch more of these more of these I got them off 25 bucks already sold one set of knives here there was Bush Cheney and a young a teenage boy bottom he's probably 19 maybe 20 he he bought it and that one kid says well Trump's running for president he said yeah but these are collectors and he collects knives like this so I sold it for 10 bucks got $10 of the 25 I spent on more products it's already got that like I said there's there's 63 dollars worth of stuff in alone just this these are five these are two dollars beast that's 10 and I got all that stuff and five picture frames and a couple other things I got for $25 so I already got 10 of that back that is there's 63 dollars worth of stuff merchandise right in there right now I'll sell a whole thing for 30 so yeah I got wiggle room on my stuff so anyway so that's what I'm doing just doing a little vlogging and selling hopefully I don't get skunked because I didn't bring a lot of stuff more stuff you got more chance you got a pupil stopping so yeah I'll let you know how I do as the day progresses monkeys got a short day so she'll be here about two o'clock or something I like to stay down here till about four there are some clouds rolling in it it's pretty nice day but it's not supposed to rain it's a little chilly but it should warm up so I'll let you know what's happening okay guys so nothing exciting happened same old same old I got skunked for the very first time I sold nothing nothing nada didn't sell anything didn't make a dime but I'll show you what I'm doing tonight what you probably won't be able to see this video till tomorrow because it's going to be a while and then it takes forever to upload so you probably see it tomorrow but it'll be the video after this one boil peanuts my own recipe I called my own recipe because I've never made peanuts before so we'll see how it turns out I'm gonna try to sell some of them everybody's selling boiled peanuts so I've got I've got my cups they usually sell these cups 18 ounces for three three dollars and small it's two dollars so I'm gonna sell these at two dollars for they usually sell these six 16 ounces they usually sell 16 ounce cups for three bucks I'm gonna sell them for two and I'll put them in I got plastic paper bags over there you put them in paper bag in the cup and then in the paper bag so I'm gonna try to sell them and I'll let you know how they turn out but so anyway stay tuned for that bit video it's going to be under chef Shebaire playlist so you guys might be interested in that some of you especially southern people and so anyway I'm making Cajun by the way it's not just regular salted peanuts I'm making Cajun style so it's gonna be spicy and they should be good all right so anyway guys thanks for watching appreciate it monkeys in there watching one of my videos she didn't get to see yet that uploaded last night so alright guys Shebaire the myth man legend gone from now by the way happy new year guys stay safe be cool and remember Shebaire 2020 for president


Hey, what's good y'all should boy down from the dawn showing friends and we packed for the decade pop reaction I'll check this out. We're gonna have in hyphen Paradox invasion dance practice believe it came about four days ago So we're gonna get ready to y'all make sure you hit the like button make sure you ring that bell share subscribe and you know Show love to the page and also, please check my description I will be I have a fundraiser right now for my dance competition that I'm hosting for the dance community So if you would like to support for my dance competition for the community Check the link in the bio. Well, link in the description, but other than that y'all let's get into it and three two one All right, let's see what we got y'all and hyphen been on work right now if they've been working Okay Oh got a hip hop feel I know when I see that move is I'm seeing some hip hop Did you see that formation transition? That was crazy. I like that That's a yeah This looks so bow bow bow.  Bye. Bye. Nice isolation with the head Oh, I ain't gonna lie when you got those part where you gotta be like what what what or anything or any type of song and you make choreography sometimes Those are the most easiest things to mess up on because you will forget your little small part right there So shout out to them getting that done for this video Ah let's go See what they do with this you can tell that they're like real mellow right now with their movements because you mean Everybody great snaps on Hey, okay. Yo shout out to the music tricking me. I thought it was about to go higher And it gave me a real mellow I like that Two three and four Hey, go back. Watch their feet though. Watch their feet. Watch their feet So when they do that jump with the arms, I'm gonna put this water down. Y'all know I keep my water When he put his arms up his feet said boop boop boop boop Did you see it? You see that? How he was doing that with his feet It's just the little things for me like how they how he was moving too Like he was just moving up just a little bit more and then the rest of the group was coming in Oh Hey, okay, let's go listen I'm about to mess with y'all That's the part when When y'all in the concert y'all on a tour y'all see that that's where y'all gonna go crazy.  No, I'm not. I didn't see what they did you saw I'ma play for you one more time Because dance teachers and choreographers, we will put this in a piece anytime to get the crowd screaming Just so you know It's coming watch Yeah, look at y'all look at y'all. Oh my god. Yeah, stop.  Hey Very very Very social dance hip hop type of choreography very very really big is like the moves are very familiar for me So it's like I'm not really seeing any new moves that I'm not used to But one thing I'm loving about what they're doing with this movement because what I mean by that is There's certain things that the og's and then hip hop culture teaches us And for sure one thing I've been knowing about kpop these groups take class and they train They absolutely train. Hold on. Let me put my phone on silent because people want to talk to me when I'm talking to y'all Back to the video so But the way they put it together in this choreography like this whole transition here was cool Ah, okay. Y'all seen that before but watch how they put it together Yeah, they put it together very well how they're moving with it. Let's go Come on. Okay. Yeah, you gotta groove that you gotta groove that Okay Man, what if the beat would have had a bang right there when he went back One two bow boom boom. I like that. Okay Very very got a lot of jumps and bounces.  I know y'all went crazy I want to see y'all reaction went in when concert doing this y'all gonna probably go like How many steps that he doing there five three four steps Okay, sorry having a dance moment. I mean steps. What's that right there? I mean steps with that Two five five five five five five five three four steps five six seven eight one Yes Really reminds me. Yeah so much of yeah. Yeah, it reminds me of like the The 90s of hip hop choreography I haven't said that in a long time. Actually, I think it was last year when I got into kpop And I saw a lot of the videos that had a 90s flavor of hip hop And what I mean 90s to be correct with you. Let's go 93 94 Early 95 Because I remember that time Yeah, let's go paradox Top top top five five five one and two and three. Yeah. Yeah I kind of felt that it was gonna leave that that line Yeah And back in that back in that formation Very well put together choreography got right back into this. Yeah Yeah Yes So there's movements when it comes to hip hop and when it comes to any dance style honestly that you probably know Or you love watching there's a familiar with movements that you've seen before But now it's going to be about how well can you put it together? And that's what i'm trying to explain is if if me you okay audience If me you right whoever's watching this we both go take uh, let's go with um popping If me you both go to a popping class or Let's say we go to a crumb class. We both learn the same basics We learn transitions and all Now as we learn and grow in our experience with the dance Now i mean you meet up face to face in a battle or showcase or anything It's going to really be about who can say it the best with this movements And the movement becomes a language to two people when they watch it And they're taking that movement that they learned to study And now they're making a choreography and they put it together very well because a lot of people have the same movement It's them like the bounce the dance style right and can't put choreography together for none in the world Seriously seen it witnessed it da da da Yes Good slides. Come on y'all. I like in hyping slide Ah Yeah Yeah, yeah Good work. Good work. Good work. Let me show Down right here. Good work. Good work. Good work. Good work. Good work. Definitely enjoy that for me It's like right now i'm starting to see more in hyping show up more So i'm just kind of like i'm in the middle of that journey watching them and i'm like, okay. Keep it coming It's like, okay. This is good. This is always y'all always been good because i'm i'm the rookie to in hyping I don't know much about them. So i i'm going on this journey now Learning more about in hyping i'm like, okay. Keep it coming because i'm loving what i'm seeing and you're starting to see So much diversity in their in their in their music that i'm listening to and then also watching the dancing and y'all know As me what i do i watch dance So i'm starting i'm always starting to see more and more of them that lose and now they're showing this hip hop side of them That's really crazy with the bounce and the rocks and the boogie and the flow And they really took those social dances of hip hop and really getting into that movement All of that and the jumps all of that. Yeah. Yeah And you've seen them know a dance style in the culture. You know what i'm saying because hip hop is a culture And so with that They took that flavor right that context And boom they smack y'all with that Shout out to them But listen y'all if you enjoyed this video, please hit the like button share I think y'all so much are showing me love and i will see y'all soon Peace