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Suppose A is a 2 X 2 matrix whose entries are real numbers; and suppose A has 2 eigenvalue 1 i with corresponding eigenvector ~iWhich of the following must be true?...

Question

Suppose A is a 2 X 2 matrix whose entries are real numbers; and suppose A has 2 eigenvalue 1 i with corresponding eigenvector ~iWhich of the following must be true?2Ahas eigenvalue 1 + i with eigenvector(1+1) 2 has eigenvalue 1 with eigenvector (+4) ( 2 has eigenvalue 1 + i with eigenvector 1 _ i-2 has eigenvalue 1 + : with eigenvector -1+i

Suppose A is a 2 X 2 matrix whose entries are real numbers; and suppose A has 2 eigenvalue 1 i with corresponding eigenvector ~i Which of the following must be true? 2 Ahas eigenvalue 1 + i with eigenvector (1+1) 2 has eigenvalue 1 with eigenvector (+4) ( 2 has eigenvalue 1 + i with eigenvector 1 _ i -2 has eigenvalue 1 + : with eigenvector -1+i



Answers

The $2 \times 2$ real symmetric matrix $A$ has two distinct eigenvalues $\lambda_{1}$ and $\lambda_{2} .$ If $\mathbf{v}_{1}=(1,2)$ is an eigenvector of $A$ corresponding to the eigenvalue $\lambda_{1},$ determine an eigenvector corresponding to $\lambda_{2}$.

The topic of this question is Eigen values and I convicted. This question asks us to show that this vector is an Eigen vector of this matrix and to find the corresponding island valley. Recall that vector X. Is called the or an Eigen vector of a matrix A. If a times X is a skill in multiple effects, so to show that this is an Eigen vector, we have to show that this matrix times inspector is this killer multiple of the vector some scale, there are times 111 And we can even say if we call this killer lambda, we can call this lambda lambda lambda. So we can work out this product since the dot product. Ever since this first entry is given by the dot product of the first row and the first column, the entry will be given by two times one plus negative one, times on this negative one times one, so two minus one minus one. Similarly the second row uh dotted with this, this column vector, we'll give the second entry. And since these are all ones, they don't really change needs when you take the dog product. So the result will be minus one plus two minus one. And similarly the last road will also be to negative ones added with a two attitude to. And so since we have zero 00 we know that our land A is zero zero times 111000 And in fact zero times any vector. Is there any three components factor is 000 So we know that this collector is an Eigen vector and its Eigen value, its corresponding on your value is zero.

We can see that the product in this case is one times one plus two times minus one on the second. Entry is going to be three times one plus two times minus one on this is the same, uh, minus one on one. On notice, we can write this. ASUs minus one times. Victor one and minus one. Okay, so we had eight times X equals minus one times X, which implies that exists an icon, Victor. I am Victor on the corresponding I am. Value is my nose one.

We're asked to find a real two x 2 symmetric matrix a. With certain Eigen values in part they were told it has Eigen values, lambda equals one and lambda equals four and an Eigen vector U equals 11 belonging to the Eigen value. Lambda equals one pressing you. Well, if A is a real two by two symmetric matrix, then A has the form little A B b D. Now we know that the characteristic polynomial of delta T. This is t squared minus trace of A. Which is A plus D, M. T plus the determinant of a. Which is yeah, A d minus B squared except been sleeping my houses thing And we want λ one in λ 4 to be zeroes. Yeah, So we have delta of one. Well this is one minus A plus D plus a d minus b squared and this is equal to zero. Likewise delta of four yes is equal to four square which is 16 minus four times a plus de plus a d minus b squared equals zero. Good lt so we have two equations and three unknowns can simplify by subtracting the second equation from the first equation. This way we eliminate B so we get one minus 16 is negative 15, negative one minus negative for is plus three tens. A place de equals zero. We work was already so we have that A plus D. He's going to be equal to five. Right? Oh Now on the other hand, we know the Eigen vector 1. 1 of λ one. This means that uh a times U. Is equal to unit or one times you? Great, first of all, no second rate D equals five minus A. And so we have a B B five minus a times the vector 1, 1 equals the vector 1, 1 again. So we have a plus B right in b plus five minus a. So five minus a plus B Equals The Vector 1 1. And so we have a system of two equations in two unknowns adding the first to the second. We get five plus two B equals two. And therefore B is equal to negative three halves are really and therefore A. Is equal to one minus B. So one minus negative three halves which is 5/2. And so a. is the matrix five halves -3/2 -3/2. And then going back to calculate D. D. Was five minus A. So that D. Is five minus 5/2 which is 5/2. So this is five halves negative three halves negative three halves five halves again. Yeah so that's the answer for part A. Some power six. Oh then in part B. Well I'm sorry we're also asked to find a matrix B. For which B squared equals A. Yes. Mm. Well to do this, we want to diagonal eyes A. Yeah, dumb bitch alert. Yeah someone right right diagonal eyes. A need to find an Eigen vector associated with the lambda equals four. Dumb bitch live. So the matrix mm This is a. Yeah. About how minus four. I. Oh they yeah they made it look like the fucking because you know they open the chest and this is the matrix boy 5/2 -4. It's -3/2ves negative three halves negative three halves made of three halves which corresponds to the system X plus Y equals zero. And so another Eigen vector is you get there uh negative one. The spread they opened the chest. So well Megan vectors V equals one negative one. Perhaps. What important? Oh now it's like some new style shit open. They got all the same rest. Okay so we have this Eigen vector. Yes. Have you now since a a symmetric or by direct calculation it follows that U. And V. Are orthogonal. Yeah. Shit. And you have the unit factor associated with you is ah One of the route to one of route two. And the Eigen vector associated with the hat is one of her two times one negative one. Yes. Yeah. Yes. And they have a carvell. Mhm. Okay. You can the car. Um And if we take P. To be the matrix with columns, you had the hat? So 1/2, 1 over to uh 1/2 and negative 1/2. Sure. Then it follows that A. Is diagonal. Izabal just and we can write A as P inverse times. Sorry, we can write D as p inverse ap Which is the Matrix 1. 4 diagonal. Those things are so. Yeah. And therefore rearranging if B squared equals A, then it follows that B squared is equal to um P times D. Ten's PM verse. And therefore B is going to be p times the square root quote unquote of D. T. P inverse, which this is going to be Phil p notice that p is also a orthogonal matrix. And so this is one over root two Times, 1, 1 1 -1 times squared of D. Which is square diagonal entries. 1002 times PM verse. Which foreign flag? A matrix is the same as Pete transpose, which is uh the same as P. Actually this is going to be times one over root two times 111 negative one. Because I can't and not fucking trying to figure out. And this is equal to 1/2 times. And this is Uh 3 -1 -1. 3 shoot local this all the time. I can't stop fucking doing it. And then it gets even like doesn't do the hair to stop growing. It's a constant chewing on it like a fucking cow. Yeah. Which is of course 3/2ves negative. One half negative one half three halves going out. This is the square root quote unquote of A. Yeah. Likewise for part B. We're told that we have again values lambda equals two. Lambda equals three. And I can vector u. equals 1. 2 belonging to the Eigen value. Lambda equals two. Yeah. So following a similar procedure to what we did in part A. We find the characteristic polynomial of a general matrix two by two matrix A. So no I think then we plug in our Eigen values two and three. And we use this to derive a relation between A. And D. Here's to then we use the fact that you an Eigen vector so that A. U. Equals two times you to find relationship between Adam says he's A. And B. After substituting for D. Well, well lucchese who And from all this information, you should eventually find that A. is equal to 14/5 B. Is equal to negative 2/5 things. C. Is also we'll see is the same as B. And d. is equal to 11/5 Woods waxing. What do you think about waxing shit out of your ass? And and therefore the matrix A. Is of the form 14/5 -2/5 -2/5 by symmetry and 11/5. Yeah, new york. All right. This is a symmetric matrix with these again, values. Now we want to find a matrix B. For which B squared equals A. Once again to do this, we want to find the orthogonal decomposition of A. He was like, We already have one. I can victor you let's find an eye, convicted. Be associated with lambda equals three. Okay well at your your dick. We solve the homogeneous system associated with m equals a -3 times I. And you should get Yes, absolutely. Yes. You're not a real mail tribe. The Eigen vector. Ask him the yeah you're allowed into the arm. Yeah. Yeah. See two negative one is in fact an Eigen vector belonging to I can value land to which is three white. I went to home then we find the unit vectors you had. And the hat So you had is one over root five times 12 and V hat is one over root five times two negative one. Then once again since A. Is symmetric, it follows that U. And V. are orthogonal really. And we have that A. Is diagonal. Izabal. Mhm. Two since we have two orthogonal Eigen vectors and it's harder our change of basis matrix P as columns. You had the hat. So uh one of the route five, two of the route five and two of every five -1 of a route five jake. And we have that the matrix D. Which is p inverse times eight times P. Walk. This is the matrix whose diagonal entries are the Eigen values two and 3. Shoot your way with which I this is diagonal, I would say. And therefore if B squared equals A then it follows that this is also equal to P. D p inverse. Therefore be is going to be equal to p times the quote, unquote square root of D times p inverse. Which noticed that P is in orthogonal matrix as well. Since it was symmetric or a with symmetric. And this is P. Which is one of the route five Times The Matrix 1-2 -1 issue. See that's for times squared of D. Well this is the matrix whose diagonal entries at the square roots for the diagonals. Route 200 Route three and PM verse. Which is just the transpose of P. This is one over root five times 1- 2 -1. Mhm. Best you rock. If you carry out the matrix multiplication. This in turn is you get the matrix. Let's see. Route two Plus 4 three. Who knows what kind of a cult shit? Over five. The Chavez. Right? Didn't get like some crazy for cancer. I where all I know is before that guy. He's one of the you know, speeches with tracksuits. I didn't I messed up. Yeah. Our The next entry is to route to -2 or three. President over five. The next entry is two or 2 -2 or three. Again, they went on sale, discontinued over five And finally forward to plus route three track clinton check. Right now, we're over five. Yeah. This is the square root of our matrix A. I shouldn't.

This question covers topics relating to the linear and running system of linear equations. Okay, But we will start from forming a new matrix. Vice of attracting lander for the from the organo interests of a given matrix. Right? So if we do you do that. You are going to get two minus lambda one and one negative lambda minus two. Okay. Right. And then you are going to find determinant of the obtained matrix. Right? So determinant of this matrix and they noted by a minor lumber I write. So this name is, it's called a a man Atlanta. I is going to be λ Square -5. Right? And then you solve the quadratic equation and then you get to judge right, London one equal to negative credit. Fine and London two equals the square root of five. So these are the Eigen values right? These ah I can values Okay, next, we are going to find the Eigen vector. Right? All right. So how do we do that? So for first I got valued the negative square to the five. You should lock in to have the matrix. Right? So the matrix is two plus quarter to five. One and one negative two plus quarter to five. Right? And you compute a noon space of this matrix and the noon of this matrix is So after computation, you are going to get this right And similarly you do for the second Eigen value. And finally you will get the new the new spy of the agon Matric? S the after two plus square root of 5 1. That's it


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