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 λ^{2}  3λ + 2 = 0. But then I couldn't get a matrix with positive entries to work for that. 
Re: Linear Algebra  Matrix with given eigenvalues
Pick a diagonal matrix.

Re: Linear Algebra  Matrix with given eigenvalues
Does that count for the entries being positive though?

Re: Linear Algebra  Matrix with given eigenvalues
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Re: Linear Algebra  Matrix with given eigenvalues
thanks though!

Re: Linear Algebra  Matrix with given eigenvalues
The 2x2case is not so difficult. Remember (or prove) that the characteristic polynomail of a 2x2matrix A is
[tex]\lambda^2tr(A)\lambda+det(A)[/tex] By the way, I think your characteristic polynomial is wrong. 
Re: Linear Algebra  Matrix with given eigenvalues
??? Why do the diagonal matrices
[tex]\begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}[/tex] and [tex]\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{bmatrix}[/tex] NOT count as "all entries postive"? 
Re: Linear Algebra  Matrix with given eigenvalues
He probably doesn't consider 0 to be positive.

Re: Linear Algebra  Matrix with given eigenvalues
But it is much easier to claim that 0 is positive!:tongue:
Thanks. 
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 i1 3 And I did get the right eigenvalues from that. Does that work? 
Re: Linear Algebra  Matrix with given eigenvalues
You still have 0 as an entry, you don't want that.

Re: Linear Algebra  Matrix with given eigenvalues
Yeah, I didn't realize that at first. :/

Re: Linear Algebra  Matrix with given eigenvalues
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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.

Re: Linear Algebra  Matrix with given eigenvalues
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Re: Linear Algebra  Matrix with given eigenvalues
Well, any value of x between 1 and 2 (like 1.1) work.

Re: Linear Algebra  Matrix with given eigenvalues
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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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