1) Let1 0 2 A 0 2 0 2 0 Find the eigenvalues of A Find nonsingular matrix P such that P-1AP diagonal Find D without computing P-1D is...

$1-8$ Show that $A=P D P^{-1},$ where $P$ is a matrix whose columns are the eigenvectors of $A,$ and $D$ is a diagonal matrix with the corresponding eigenvalues.
$A=\left[ \begin{array}{ll}{2} & {0} \\ {0} & {1}\end{array}\right]$

Mhm. Were given a matrix A. She might. Mhm. A. is the two x 2. Matrix 2 -1 negative. 23 Let me ask this. You see some copies. Okay, in part A we're asked to find the ideal values and corresponding Eigen vectors of this matrix do this. Let's find the characteristic polynomial for a two by two polynomial. This is two by two. Matrix. This is T squared minus The trees today, which is five times T plus the determinant today, which is eight. Uh huh. And we can factor this as Uh huh. She said was like Hold on 1 2nd. Yeah. So the problem with the mic. Yeah. Moving on. Just see. Just hold now. So there should be instead of plus eight plus four. And so this can actually factored as T -1 times T -4. Okay. And therefore the zeros of the characteristic polynomial are Eigen values. Lander one equals 1 And landed two equals 4. His name is Neil Israel. Mr ross. What's wrong? Yeah, we'll begin with the first dragon value. Lambda One equals one. I'll subtract one down the diagonal. Day we get the matrix M. Which is a minus I. This is the two x 2. Matrix 1, -1 -2. Okay yeah. Two. This corresponds to the modernist system, X minus Y equals zero and negative two. X plus two. Y equals zero. And this corresponds to the single equation, X -Y equals zero. So take Y equals one, then excess equal one. There's only one independent Eigen vector through. I'll call it you so that you equals 1. 1 is an Eigen vector against the belonging to the Eigen value lambda one which is one. Jason's Shaller Jared. Uh huh. Jacob Haines and Christopher eight now mm Their second Eigen value four. I'm going to subtract forward down the diagonal of a. So we get our matrix M. Which is a -4 i. And this is the two x 2 matrix negative too -1 -2 -1. This this gives us the homogeneous system negative two. X minus Y equals zero and negative two, X minus Y equals zero. Which is of course just a single equation negative two, X minus Y equals zero. My question is for And if you take yeah. Bodies. Mm Yeah, X. To be positive one by is negative two. And so we get the vector V which is one negative two. And this is an Eigen vector belonging to the Eigen value Lamba to which was four. His car. Mm hmm. Okay. Then in part B for us to find a non singular matrix P. Such that matrix D which is p M a p is diagonal. Yeah. Well, because we have a maximal set of the nearly independent Eigen vectors which is U. And V. For 11 And 1 -2. That would be cool if we three men. Yeah, sure. But no, I wouldn't raise a child with you guys. Sorry. Uh Just one baby. I wouldn't. So it follows that the matrix P whose columns are these hiding victor's is non singular and diagonal eyes is our matrix A. So we'll take P. To be the matrix with columns 11 And 1 -2. It was hopefully before. And as a result we have that is diagonal Izabal and D. Is equal to p inverse Ap where D. Is diagonal matrix of corresponding Eigen values for entries. So 1004. So, Mhm. Then in part C. Mike, we're as to find A. To the eighth power and find F where F F. T. Is a certain polynomial. What would you do Personal? Find a to the 8th power. Pretty upset. Well, we'll actually use a result from part B. A. To the eighth power rearranging. This is P times D 10 P inverse to the eighth power. Which of course is p tens deviate power times p inverse. So a few things first. Mm We know that uh D. D. A. Is the same as D. But the elements are raised to the eighth power. So this is going to be The Matrix. 1. 1 1 -2. Can I read times Matrix one of the eighth, which is 100 and 4 to the 8th, which is 65,000 536. Are you just look this is his name. He was going through shows be funny. I see. And then we multiply by the inverse of T. Well, he is two x 2. Matrix. It's fairly easy to find the inverse. It's one over the determinant of P which is one over negative three negative one third times the matrix while we swap. They do we have negative too negative one negative one one sort of which is if you distribute the negative one thirds uh positive two thirds. Although we can really leave the negative one third out. More helpful in an elixir with which you can disrupt society kind of like true. You know where they want to control people with dangerous drugs. Do you carry out multiplication? You'll eventually get the matrix 21846. We should yes, -21845. It's true, -43690 Yes. and 43006 91 child. Yeah. Which is what he did. And they went to France. So this is 88. Okay. You we can't get their tragedy. Now the matrix we were given F. Sorry. The polynomial F. Is F. F. T. Equals T. to the 4th -5 T cubed plus seven T squared minus two T. Plus five. Once again find FJ. Sure. Well, rather than find it directly, just plugging in. Notice that we can replace a by. Yeah P. D. P inverse. So this is P. D. P inverse to the. Well actually if you want to apply the methods this is the same as P times F. Of D. Time is p inverse? Probably 60. Yeah. Now if you d is a diagonal matrix and she's because she was always quote unquote a fat bitch. What are you floating there? You're his little jerk. And so this is going to be uh P times Well, matrix with entries is in the new um As of it said half of one fat bitch. 00 F of four times p inverse. Now we'll plug in values. So we see that F of one plugging this in this is one minus five is negative. Four plus seven is positive, three minus two is positive, one plus five is six. Yeah. And f. of four. Let's see that makes sure 4 to the 4th I guess minus five times one. Oh 4 to the 3rd. Yeah. And I'm not even being credit. I just want to know plus seven four squared exactly. Yeah. As a matter of scientific interest -2 times four Plus 5. Is it either? I just it's interesting to know how things change sides, know the number. Oh it's Yeah. Which is 45 being like that's that's flashy there And therefore FV is also equal to what we found. P was 111 -2. Let's see. Sure Times The Matrix 600. 45 times P inverse. Which we said was negative one third. This is a what Times. -2 -1 -11. Okay. Sense if you carry out the matrix multiplication is here you think about big to make we get that F. Of a is equal to the matrix three. Just because you can't dick fuel one two one. We were jacking off to should we do another? Mm. Yeah. Questions for finally in part D were asked to find a matrix B such that B squared equals a. One would say this is like a square root. Uh A super verbally. Well, if B squared is equal to a this means B squared is equal to p times D times PM verse shoot. Therefore B will be equal to p times D two P inverse, that one half, which is the same as a P times D. To the one half times P. And Bruce is right. But in the sense that the the one half would be a square root of C pound chip. Challenge that we get a powerful shit Now to find the square root of D diagonal matrix. Who just square with the diagonal entries. And so this is going to be the matrix P. Which is uh 1, 1 1 -2, think coming times the square root of D. Which is the square root of one is 100 In the positive square root of 45. So plus size. No, the crab if there was a black Yes, here we go. It was like I got a tanning booth for my apartment. I'll see you guys in two years. You'd be like, I got to see how dark this month. Mhm. Okay. These are the square of four which is to times p inverse which is again negative one third times the matrix negative to negative one negative 11 hell covered to black. And eventually this simplifies to the Matrix 4/3. Um So you would say -1 3rd and negative 2/3 five thirds mirror. You know the globe? Imagine how much more disrespectful that is to Russia. Okay. First.

We have the matrix. Hey equals one 01 210 301 And we want to show that this matrix is similar to a diagram matrix, meaning that this matrix equals P D p inverse where d is a diagonal matrix with its with the diagonal entries being the Eigen values of A and P being a change of basis matrix where the columns are the itin vectors of a well, if a is similar to a dia go matrix, then this will be true meaning that if so long as d is diagonal and we have this Abel's pdp inverse then d will have I in values on I you know surveillance. Dagnall and P will have its high in vectors as columns. So all we have to do to show that it is similar to a diagonal matrix is show that it is Dagnall Izabal. Now to show that is diagonal Izabal we can start by finding its wagon values by taking the determinant of it with it's attracting Lambda on the diagonal to take this determined I'm going to expand with respect to the second column cause as two zeros in it and what do I get? I get one term with coefficient one minus lambda in the negative sign because by the checkerboard matrix, this is goingto have ah, negative sign attached to it. And I cross this out their own column that it's in and I take the determined one minus lambda, uh, squared minus three. And we can. What you can do now is because this is not a factored polynomial. You can. You can't just instantly read off the roots as Lambda equals one, for example, because, well, that's not a root of this. But if you do expand this, then this will be a cubic polynomial. In Lambda, there is a cubic formula and so you can plug into the cubic formula. And if you plug that in, I'm not going to do it here. Not gonna write it out because it's quite long. You plug it into the cubic formula, you will see that all the eye in values here are actually distinct, which means that we have three distinct Eigen values, which means that we have three associated linearly independent, high in vectors. And if you have three linearly independent Eigen vectors, you can write a basis for our three. Our space that the Matrix is acting on. How do these Aiken vectors? And if your basis of Aiken vectors, that means that your matrix is similar to a diagonal matrix?