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. = __ 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=====conj(a)
Dividing by 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 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 = 0 for all v and w, in which case every matrix is
Hermitian, whether the eigenvalues are real or not.
dave
__