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Points) Find the eigenvalues ^, 12 and associated unit eigenvectors U1, U2 of the symmetric matrix-17 A = 9|. 9The smaller eigenvalue 1110has associated unit eigenv...

Question

Points) Find the eigenvalues ^, 12 and associated unit eigenvectors U1, U2 of the symmetric matrix-17 A = 9|. 9The smaller eigenvalue 1110has associated unit eigenvector U]The larger eigenvalue 12-20has associated unit eigenvector U2Note: The eigenvectors above form an orthonormal eigenbasis for A_

points) Find the eigenvalues ^, 12 and associated unit eigenvectors U1, U2 of the symmetric matrix -17 A = 9|. 9 The smaller eigenvalue 11 10 has associated unit eigenvector U] The larger eigenvalue 12 -20 has associated unit eigenvector U2 Note: The eigenvectors above form an orthonormal eigenbasis for A_



Answers

Given that $\mathbf{v}_{1}=(-2,1)$ and $\mathbf{v}_{2}=(1,1)$ are eigenvectors of $$A=\left[\begin{array}{rr}-5 & 2 \\1 & -4\end{array}\right]$$ determine the eigenvalues of $A.$

This problem asks ist friend Eigen. Values of the Matrix A with the given Eigen vectors could do this using the equation. A times V is equal to Lambda Times e So first for the first Eigen vector V one, you can plug in a 4213 then V one vertically. One negative too is equal to lambda Warn times one negative, too. Then doing matrix multiplication on the left side, we got the top row to be four times one, which is four plus one times negative two, which is negative two and two times one which is two and three times negative two, which is negative. Six is equal to to negative four and that is equal toe lambda one times one negative too. So we could see the constant Lambda 11 Dividing to divide by one and negative four by the bynegative two gives us land. One is equal to two. Then for the second Eigen vector view, too, we can plug in again a 4213 Then the vector V two wishes 11 is equivalent to lambda two times 11 again matrix multiplication of the left side, The top row. We get four times what it's for. Last one times one, which is one 1212 plus three times one is three, which gives us 55 which is equivalent to land a to times 11 that if we divide five Vector 55 by the Vector 11 We find that landed too is equal to five, and those are two Eigen values.

This problem asked us to find Ivan values and Aiken vectors of given matrix. You can do this by funding the characteristic polynomial, which this inviting the determinant of a minus land of times. I That would be the determinant of 10 minus lambda zero negative. Eight. Native 12 to minus lambda 12 80 96 minus Landau. Okay, solving this out. We get 10 minus lander times two months. Slammed other times a night of six months. Lambda minus eight times, two months. Lander times negative eight. And then the other two for, um, the subtraction are zero spoke due to the zeros in the Matrix Southern. This out a bit more we can get. Um well, we find that we get negative 60 months for Lambda Plus Lambda Square plus 64 0 sorry. We need to include the two months slammed up minus it becomes a plus 64 to minus slammed up, which simplifies to because you can add the 64 in yet for minus four lambda plus lander squared times two minus lambda, which is equivalent to tu minus lander. Cute. Then with that, that means that our Eigen value is two with an algebraic multiplicity of, um three. So this means that the Matrix could have or this AG value could have three Eigen vectors associated to it, but a maximum of three. So solving for the Eigen vectors, we confined this by finding anyone slammed I and the final director X that gives the zero vector so plugging in to, yeah, 10 minutes to wanna leave blanks for zeros for save time. Negative. Eight. Um native 12 to minus two 12 and then eight. Negative. Six mines, too. Times the Eigen Vector X that is equivalent to eight negative eight and 12 0 12 Negative eight Over here. Eight. When we add the zeros to this room, then with gash in elimination, if you add the top row to the bottom row, that gives us eight negative 12 8 Then zeros on both these bottom row's times X, which is equivalent to if you divide by four. That gives us two negative 32 all zeros against here times the vector A B C shaped equal 000 and without weekend fronts literally Independent solutions, ABC to the system of Earth to the Single equation to AG minus three b plus two C is equal to zero and she saw this. You'll find that the two linearly independent ah solutions you'll find from this are going to be like 101 and three 20 meaning that this adding value has geometric multiplicity of two. So this is our final solution for Agon vectors and our Ivan value is what it to

We want to find an Ivan Vector of the Matrix 9023 with wagon value. Nine. Well, let's say our factories X y. On one hand, this equals nine x and two x plus three. Why, on the other hand, because the night in Vector with Ivan value nine. This equals nine x and nine. Why so nine X equals nine. That's no matter what X is. And we have two x plus three. Why equals nine. Why? Which we can simplify to two. X equals six. Why and X equals three. Why or better yet, why equals X over three. So when I can vectors anything is any vector which has any value of X and 1/3 that value of Why So, for example, 31 would be such a nag factor. No. For the next Matrix, we do the same thing we have 1257 We wanted Ivan Vector that has an icon value of four plus square root of 19. So we have X. Why on one hand this equals X plus five. Why and two X plus seven. Why? And on the other hand, this equals work whispered of 19 x times for four plus squared of 19. Why so we have two equations. X plus five. Why equals four plus swear it of 19 x and X two x plus seven Why equals four plus squared of 19? Why? So what can we do? What would be the easiest way to solve this? Well, let's say Let's solve this first equation for Let's actually solve the second equation for X. That seems to be the simplest thing. So we have X equals four plus squared of 19 minus seven times. Why divided by two, which simplifies to squared of 19 minus three. Why over two? And plug that back into here. So we have, I swear it of 19 minus three. Why all of that over to Plus five. Why equals four plus squared of 19 and then multiplied by squared of 19 minus three all over, too. Why so no, we can isolate why we can factor out why and actually get why multiplied by squared of 19 minus three over two minus five minus four plus squared of 19 squared of 19 minus three all of that over too. And in fact, this equal zero, which means that Why? Well, let's see. Does this equal zero? If this factor is not zero, then why must be zero? So let's take a look at this factor. It's squared of 19 minus three and I want to write everything over to someone to make the minus five minus 10 and then minus for swear it of 19 plus 19 minus. I'm sorry. That should be a minus 19 and then plus three squared of 19 and finally plus 12. So all of this over to now this numerator simplifies to wave squared of 19 minus four, squared of 19 plus three squared of 19. So those actually canceled and we have minus three minus 10 is minus 13 plus 12 is minus one minus 19 is minus 20. Divided by two is actually minus 10. So we do. In fact, get that Why equal zero? And if y equals zero, then we have that to X equals zero. And, uh, from this equation, and if two X equals here, then X equals zero, and the vector 00 is not considered an Eigen vector. So we don't have an I in vector for this value. Next, we have 103200 0 to 1, with bye in Vector, which has Aiken Value. Three. Let's do X y zem. It gets us X plus three z two x and two X plus z should equal three x three Why three z We have X plus three z equals three x two x equals three Why and two x plus z equals three z So from this first equation, we can simplify thio three z equals two x or X equals three halves Z um, from the second equation, we have X equals three halves. Why? So we in fact have three halves? Why equals three have Z which means that why equals e And so now in this 30 equation, we can substitute in X three halves Z for X so I get three z plus Z equals four z So we get four z Abel's for zine. So what we get is that Z can be any value. Why has to equal to Z that X has to equal to three have busy and any Eigen vector of this form. We'll work the next one 0312 We want Aiken value of minus one. So again. We do the same thing I get. Why three X plus two Why? Which should equal minus X minus y. So you get why equals minus x three X plus two y equals minus y so I can substitute in minus X for y. In the second equation, I get three X minus two X equals X. Yeah, so X, as we see can be This gets X equals X so acts could be any value. And why just has to be the negative Any vector of this form will satisfy will be an I in vector with wagon value minus one. Next we have 0111 times x y with Well, let's first write it out as why X plus y. And this should equal one plus square to five over too. X one plus square to five over two. Why? So I get why equals one plus square to five over to X and X plus y equals one plus square to five over too. Why so again I can plug in wantto plus square to five over to X in tow. Why here? And get X plus one plus square to five over too X equals one plus Where 25 over two Squared X And so what do I get? I can factor out the X and get X one minus one plus square to five over too. I'm sorry. That's a plus. Minus one plus square to five. Over too. Squared. Equal zero. So again, we need to check whether this factor is zero. If it's not zero than X, must be zero. So let's see, we have I'm gonna write everything. Well, let's write everything over four The end. So we have four plus two plus two square to five, minus one minus two square to five plus five. And this simplifies to four till two. Square to five, minus two square to five. Cancels four plus two minus one is five and then I'm sorry, this should be a minus five. So, in fact, this factor is zero, which means that ex convey anything and then why just has to be one close square to five over too X. And this is our wagon Vector or anything of this form is an I in vector and our last one. 123017 0 to 1 We wanted I can value of one plus square to 14 x y z equals X plus two y plus three z Why plus seven z to why plus z equals And I'm just gonna keep his land for now. Lambda X Landa y land Izzy, where land is one plus square to 14. So we have X plus to why plus three z equals one plus square to 14 x Why plus seven z equals one plus square to 14. Why? To why? Plus c equals one plus square to 14. Izzy. So what can we do? We can solve for why, in terms of Z in this last equation and we get that, why equals I'm gonna use Lambda instead? Could shorter land of minus one Z over too now. So we know what wise in terms of Z and so we can now substitute that into the first equation. Get Lambda minus one Z over too. Plus lambda minus one Z plus three Z equals Land X. And if we expand that where if we simplify that factor out the sea, we get Z land of minus one over too, plus lambda minus one plus three and then all of that divided by Lambda equals X. So again we can pick axe to be anything. And then I'm sorry. The opposite. We can pick we pig zine. Then why is Lambda minus one over two Z and finally X is Z times lambda minus one over two, plus Lambda minus one plus three. She's really land. Ah, plus two and all of that divided by Lambda And that's our final Agon vector.


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