From: Stefan Weigert
Subject: Hermitean matrix with prescribed characteristic polynomial?
Date: Wed, 22 Sep 1999 15:49:27 +0100
Newsgroups: sci.math.research
Hello everybody,
I am looking for a
hermitean (or symmetric) (N by N) matrix which has a
prescribed characteristic polynomial P(x) of degree N.
1. It is assumed that the given characteristic polynomial has real roots
only.
2. Clearly, there is a trival solution, namely the diagonal matrix with
the roots of P(x) as entries but this requires the explicit knowledge of
the eigenvalues.
3. The elements of the matrix should be given explicitly in terms of the
coefficients of the polynomial P(x).
4. I do not allow the diagonalization of a general (N by N) matrix in
the
course of the construction.
In other words, I look for a *hermitean/symmetric version* of the
so-called
`companion matrix C' associated with P(x). (I have not been able to find
a
similarity transform which renders C hermitean.)
Any suggestions are welcome. Thanks........
Stefan..............................
==============================================================================
From: wcw@math.psu.edu (William C Waterhouse)
Subject: Re: Hermitean matrix with prescribed characteristic polynomial?
Date: 23 Sep 1999 20:00:04 GMT
Newsgroups: sci.math.research
In article <37E8EC77.5CDC90B4@iph.unine.ch>,
Stefan Weigert writes:
>...
> I am looking for a
>
> hermitean (or symmetric) (N by N) matrix which has a
> prescribed characteristic polynomial P(x) of degree N.
>...
> I do not allow the diagonalization of a general (N by N) matrix in
> the
> course of the construction.
One solution to this problem (using only square roots and arithmetic
operations) is given in the article
MR 94i:15006 15A18
Schmeisser, Gerhard(D-ERL-MI)
A real symmetric tridiagonal matrix with a given characteristic
polynomial. (English. English summary)
Linear Algebra Appl. 193 (1993), 11--18.
William C. Waterhouse
Penn State