Eigenanalysis: Finding Eigenvectors

[erok81]

Homework Statement

This is a simple example from the book, but it gets the point across nicely.

In this problem eigenanalysis is used as a method to solve linear systems.

The matrix...

[4 2]
[3 -1]

Eigenvalues are -2, 5.

Homework Equations

(A-\lambda I)v=0

x'=[above matrix]x

The Attempt at a Solution

So I solve for the eigenvector using the first case, \lambda=-2

New matrix with above eigenvalue:

[6 2]
[3 1]

The book gives the eigenvector as [1,-3]^T

I get [-1/3,1]^T

From my rref of:
[3 1]
[0 0]

My vector and the books vector are off by a value of -3. Which I think is because I tried the rref and they just used two equations with the two unknowns and "guessed' values.

I ran into this problem a few times in the homework as well, which would eventually lead to an incorrect answer compared to the textbook.

My question is, why do you solve a normal matrix using the rref but it seems you can't do the same with eigenanalysis? Do I just break them into equations and guess solutions?

Everyone I did in the homework I ended up with the same type of answer - off by some value that I can see used to be in the matrix like the above example.

 

[Dick]

Eigenvectors are only defined up to a multiplicative constant. If Mv=lambda*v, then c*v is also an eigenvector with eigenvalue lambda. For any nonzero c. Always. Your eigenvector is (-1/3) times the books eigenvector. You are both right.

 

[erok81]

Perfect. Thank you for the quick response and the explanation.

That's good to know. I was worried since none of my answers matched the text exactly. I couldn't see why mine didn't work though - now I do. Both answers are correct.

Thanks again.

 

[erok81]

So it turns out I have an additional question regarding these problems.

This time the original matrix is
[1 2]
[2 1]

I solved for \lambda_1=-1, and \lambda_2=3

Does it matter which value you choose for _1 and _2? It seems when we found the diagonal matrices in a previous section I had to always choose the lower value in order to get the answer correct. If I didn't, my answer wasn't close due to the different matrices I could get depending on the order of lambda's I chose.

Anyway, back to this problem.

I used the above lambda values to obtain to the eigenvectors of v_1=[-1,1]^T and v_2=[1,1]^T

Since I chose the order I did, I ended up with my x_1 and x_2 for the general solution switched. This, unlike my previous, seems to matter more since they are different parts of a system. In this case, brine tanks.

Any idea where I am going wrong?

 

[Dick]

So it turns out I have an additional question regarding these problems.

This time the original matrix is
[1 2]
[2 1]

I solved for \lambda_1=-1, and \lambda_2=3

Does it matter which value you choose for _1 and _2? It seems when we found the diagonal matrices in a previous section I had to always choose the lower value in order to get the answer correct. If I didn't, my answer wasn't close due to the different matrices I could get depending on the order of lambda's I chose.

Anyway, back to this problem.

I used the above lambda values to obtain to the eigenvectors of v_1=[-1,1]^T and v_2=[1,1]^T

Since I chose the order I did, I ended up with my x_1 and x_2 for the general solution switched. This, unlike my previous, seems to matter more since they are different parts of a system. In this case, brine tanks.

Any idea where I am going wrong?

I guess I don't understand the issue. [-1,1] is the eigenvector corresponding to the eigenvalue -1 and [1,1] corresponds to 3. I don't know how the '_1' and '_2' designations have to do with the problem. You might want to quote the whole problem if it's still unclear.

 

[erok81]

I guess I don't understand the issue. [-1,1] is the eigenvector corresponding to the eigenvalue -1 and [1,1] corresponds to 3. I don't know how the '_1' and '_2' designations have to do with the problem. You might want to quote the whole problem if it's still unclear.

Sorry about that. I thought that would be an issue and forgot to go back and edit.[strike] Take a look at that post again. I've edited for clarity.

Well...I am going to after I post this. So maybe five minutes from now. :) [/strike]

The _1 and and _2 were the lambda values. But after looking at it, it doesn't matter the order of those. EXCEPT when I was doing the full eigenanalysis. The order seemed to matter there.

EDIT: You are right. I've been doing homework too long today. It doesn't matter which order I do these in, that just changes the order of my c_n*v_n answers.

My x_1(t) and x_2(t) are still swapped from what the book shows though. It looks like it might still be my same issue. My v_1 is off by a multiple of -1. i.e. If I swap the signs in my v_1, the answer comes out right.

Does that still fall into the "off by a constant" thing and my answer is still correct?

I attached a scan of my work. That should help since I am not the best at latex and posting. Sorry about the sideways scan.

 

[erok81]

To help explain, if I do it the other way I mentioned, I get this attachment. Which matches the book answer.

 

[vela]

Yes, it's just the constant factor again. You can turn your answer into the book's answer merely by replacing c₁ by -c₁.

 

[erok81]

That makes sense, thank you.

I am sure once I start doing these that have initial conditions, I won't see this problem. In fact...come to think of it, the ones I did that used initial conditions, I always ended up with an answer that matched the book.