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-   -   Linear Algebra - Matrix with given eigenvalues (http://www.physicsforums.com/showthread.php?t=593257)

 roto25 Apr3-12 11:15 PM

Linear Algebra - Matrix with given eigenvalues

1. The problem statement, all variables and given/known data
Come up with a 2 x 2 matrix with 2 and 1 as the eigenvalues. All the entries must be positive.
Then, find a 3 x 3 matrix with 1, 2, 3 as eigenvalues.

3. The attempt at a solution
I found the characteristic equation for the 2x2 would be λ2 - 3λ + 2 = 0. But then I couldn't get a matrix with positive entries to work for that.

 Dick Apr3-12 11:24 PM

Re: Linear Algebra - Matrix with given eigenvalues

Pick a diagonal matrix.

 roto25 Apr3-12 11:28 PM

Re: Linear Algebra - Matrix with given eigenvalues

Does that count for the entries being positive though?

 Dick Apr3-12 11:38 PM

Re: Linear Algebra - Matrix with given eigenvalues

Quote:
 Quote by roto25 (Post 3849064) Does that count for the entries being positive though?
Not really, no. Sorry. Better give this more thought than I gave this response.

 roto25 Apr3-12 11:39 PM

Re: Linear Algebra - Matrix with given eigenvalues

thanks though!

 micromass Apr4-12 06:12 AM

Re: Linear Algebra - Matrix with given eigenvalues

The 2x2-case is not so difficult. Remember (or prove) that the characteristic polynomail of a 2x2-matrix A is

$$\lambda^2-tr(A)\lambda+det(A)$$

By the way, I think your characteristic polynomial is wrong.

 HallsofIvy Apr4-12 06:17 AM

Re: Linear Algebra - Matrix with given eigenvalues

??? Why do the diagonal matrices
$$\begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}$$
and
$$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{bmatrix}$$
NOT count as "all entries postive"?

 micromass Apr4-12 06:18 AM

Re: Linear Algebra - Matrix with given eigenvalues

He probably doesn't consider 0 to be positive.

 HallsofIvy Apr4-12 06:22 AM

Re: Linear Algebra - Matrix with given eigenvalues

But it is much easier to claim that 0 is positive!:tongue:

Thanks.

 roto25 Apr4-12 07:42 AM

Re: Linear Algebra - Matrix with given eigenvalues

Oh, I had typed 3 instead of 2 for the characteristic polynomial. I ended up looking at this from a Hermitian matrix point of view.
And then I got the matrix:
0 i +1
i-1 3
And I did get the right eigenvalues from that. Does that work?

 micromass Apr4-12 08:09 AM

Re: Linear Algebra - Matrix with given eigenvalues

You still have 0 as an entry, you don't want that.

 roto25 Apr4-12 09:06 AM

Re: Linear Algebra - Matrix with given eigenvalues

Yeah, I didn't realize that at first. :/

 Dick Apr4-12 09:08 AM

Re: Linear Algebra - Matrix with given eigenvalues

Quote:
 Quote by roto25 (Post 3849508) Oh, I had typed 3 instead of 2 for the characteristic polynomial. I ended up looking at this from a Hermitian matrix point of view. And then I got the matrix: 0 i +1 i-1 3 And I did get the right eigenvalues from that. Does that work?
I don't think i+1 would be considered a positive number either. Stick to real entries. Your diagonal entries need to sum to 3, and their product should be greater than 2. Do you see why?

 roto25 Apr4-12 10:06 AM

Re: Linear Algebra - Matrix with given eigenvalues

Yes. Theoretically, I know what it should do. I just can't actually find the right values to do it.

 Dick Apr4-12 10:09 AM

Re: Linear Algebra - Matrix with given eigenvalues

Quote:
 Quote by roto25 (Post 3849655) Yes. Theoretically, I know what it should do. I just can't actually find the right values to do it.
Call one diagonal entry x. Then the other one must be 3-x. Can you find a positive value of x that makes x*(3-x)>2? Graph it.

 roto25 Apr4-12 10:18 AM

Re: Linear Algebra - Matrix with given eigenvalues

Well, any value of x between 1 and 2 (like 1.1) work.

 Dick Apr4-12 10:37 AM

Re: Linear Algebra - Matrix with given eigenvalues

Quote:
 Quote by roto25 (Post 3849670) Well, any value of x between 1 and 2 (like 1.1) work.
Ok, so you just need to fill in the rest of the matrix.

 roto25 Apr4-12 10:43 AM

Re: Linear Algebra - Matrix with given eigenvalues

but if I set x to be 1.1, my matrix would be
1.1 __
__ 1.9

And those two spaces have to be equivalent to 1.1*1.9 - 2, right?
because no matter what values I try, when the eigenvalues are getting closer to 1 and two, the matrix is just getting closer to the matrix of:
1 0
0 2

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