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