Hermitizing non-Hermitian eigenvalue problems

Factual Questions
1

First off, in case anyone is wondering, no, this is not for a homework problem.

Now then, suppose I have a non-Hermitian matrix M. It’ll have eigenvalues w, and right and left side eigenvectors X and Y, so that:

MX=Xw
YM=wY
YX=1

We also know that I can do a similarity transformation of M so that M’=A[sup]-1[/sup]MA; this will obviously change the eigenvectors (X’=A[sup]-1[/sup]X and Y’=YA), but won’t affect the eigenvalues.

So now suppose that I have real eigenvalues. I would think that there ought to be a similarity transformation that makes M’ Hermitian and hence X’ unitary. If there is such a transformation, how do I go about finding it?

2

Write your first equation in matrix form: M X = X W, where X is the matrix of eigenvectors (each column is an eigenvector) and W is the diagonal matrix of eigenvalues. If the eigenvectors span the space (note that this is not necessarily true for non-Hermitian matrices), X is invertible, so X[sup]-1[/sup] M X = W is a similarity transformation from M to the real and diagonal (and hence Hermitian) matrix W.

3

:smack:
Oh, of course. Given that any unitary transformation of W would give me some Hermitian matrix M" with eigenvalues W, that would explain why I can’t find a unique similarity transform to do what I want.

Lord in heaven, I’m an idiot. Thanks.

4

Nitpick

The above post by Om assumes a symmetric matrix in addition to real. If not symmetric, the transformation matrix will not necessarily be real.

But the similarity transform will still do the job albeit more elaborately.

A small note :

For real matrices, Hermitian means the same as symmetric, unitary means the same as orthogonal.

5

… Er, I’m not sure what you’re saying here. Which matrix do I assume is symmetric? The method I give will work for any non-defective matrix (one whose eigenvectors span the space) with real eigenvalues. Neither the original matrix M nor the eigenvector matrix X need to be real for the diagonalization to work. Can you elaborate on what you mean?

6

I was going to say that I have a duckie and a horsie, but I changed my mind… :slight_smile: