From: schep@math.sc.edu (Anton R. Schep)
Subject: Re: Non negative matrix with given eigenvalues
Date: Tue, 09 Jan 2001 13:23:51 -0500
Newsgroups: sci.math
Summary: Characterizing sets of reals which can be eigenvalues of positive matrices

In article <93fh9o$vc1$1@nnrp1.deja.com>, ben_geffen@my-deja.com wrote:

> If x_1,x_2, ... ,x_n are real numbers what condition they must satisfie
> in order that exists a non negaive matrix A (n by n matrix with non
> negative entries) such that the eigenvalues of A are x_1,x_2,....x_n  ?
> 
> Thanks,
> Ben
> 
> 
> Sent via Deja.com
> http://www.deja.com/

This is an old and not yet completely solved problem in linear algebra.
The conjectured condition is that x_1^k+....x_n^k >=0 for all k >=0.
Shmuel Friedlander proved that there exists a matrix A with these
eigenvalues for which a power is nonnegative. The best result is due to
Boyle and Handelman:The spectra of nonnegative matrices via symbolic
dynamics, Annals of Math 133(1991),
pp 249-316. They proved the existence of the nonegative matrix A having
the prescribed nonzero eigenvalues. The paper of Boyle and Handelman is
rather sophisticated, but Friedlander's paper is accessible (and nice to
read).
Anton R. Schep