From: rusin@math.niu.edu (Dave Rusin)
To: francstr@nic.wi.leidenuniv.nl
Subject: Re: Inner products, Hermitian matrices and eigenvalues
Date: [obliterated; must have been before  Oct. 25 1995]

In article <46lckq$o82@nic.wi.leidenuniv.nl> you write:
>Hello everybody,
>
>My problem is the following:
>I have an arbitrary inner product <.,.> on C^n, and a matrix B which is 
>Hermitian with respect to this inner product, i.e. <Au,v> = <u,Av> for 
>all vectors u,v in C^n. Does this imply that the spectrum of B is real, 
>like we have with the standard Euclidian inner product?

Here's the usual proof that a Hermitian matrix has real eigenvalues:
	Aw=a*w  for the eigenvalue  a  means
	a<w,w>=<aw,w>=<Aw,w>=<w,Aw>=<w,aw>=conj(a)<w,w>
	Dividing by <w,w> shows  a  is real.
(An inner product is assumed to be linear in the first variable, conjugate 
linear in the second). This proof works assuming that  <w,w>  is nonzero, 
which is certainly true for the usual inner product if  w  itself is nonzero.

This proof carries over to any other _positive definite_ quadratic form.
However, if you drop positive definiteness, there's no reason you can't
define  <v,w> = 0  for all  v  and  w, in which case every matrix is
Hermitian, whether the eigenvalues are real or not.

dave