Vol. 8, No. 1, 2015

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Efficient realization of nonzero spectra by polynomial matrices

Nathan McNew and Nicholas Ormes

Vol. 8 (2015), No. 1, 1–24

A theorem of Boyle and Handelman gives necessary and sufficient conditions for an n-tuple of nonzero complex numbers to be the nonzero spectrum of some matrix with nonnegative entries, but is not constructive and puts no bound on the necessary dimension of the matrix. Working with polynomial matrices, we constructively reprove this theorem in a special case, with a bound on the size of the polynomial matrix required to realize a given polynomial.

nonnegative matrices, eigenvalues, power series, nonnegative inverse eigenvalue problem
Mathematical Subject Classification 2010
Primary: 15A18, 15B48, 05C50
Received: 5 September 2011
Revised: 1 February 2014
Accepted: 5 March 2014
Published: 10 December 2014

Communicated by Kenneth S. Berenhaut
Nathan McNew
Department of Mathematics
Dartmouth College
6188 Kemeny Hall
Hanover, NH 03755
United States
Nicholas Ormes
Department of Mathematics
University of Denver
2280 South Vine Street
Denver, CO 80208
United States