Page 103 - Textos de Matemática Vol. 39

STRUCTURED MATRICES IN THE NONNEGATIVE INVERSE EIGENVALUE PROBLEM
THOMAS LAFFEY AND HELENA SˇMIGOC
Dedicated to Eduardo Marques de Sa´ on the occasion of his sixtieth birthday in recognition of his continuing brilliant contributions to matrix theory.
Abstract. Thenonnegativeinverseeigenvalueproblem(NIEP)istheproblem of determining necessary and sufficient conditions for a given list of complex numbers σ = (λ1,λ2,...,λn) to be the spectrum of a nonnegative matrix. One of the most promising attempts at solving the NIEP is to identify the spectra of certain nonnegative structured matrices with known characteristic polynomials. In the paper we give an overview of the work on companion matrices and describe some related classes of matrices. We present a method that enables us to find an estimate for the least Perron eigenvalue that we need to append to a list of complex numbers to ensure realizability and we present some examples that illustrate problems that arise while solving the NIEP.
1. Introduction
The nonnegative inverse eigenvalue problem (NIEP) is the problem of
determining necessary and sufficient conditions for list of complex numbers σ = (λ1,λ2,...,λn)
to be the spectrum of a nonnegative matrix. If a list σ is the spectrum of a nonnegative matrix A, we will say that σ is realizable and that the matrix A realizes σ.
The problem was posed by Suleimanova [22] in 1949. It has been studied in its general form [17, 8, 4, 23, 11, 12, 19, 20] as well as for the case when the list σ has real elements (RNIEP) [2, 7, 18], for the case when the matrices under consideration are symmetric (SNIEP) [21, 9, 6, 16], for the case when the matrices have trace zero [13] and for other special cases.
2000 Mathematics Subject Classification. 15A18, 15A29, 15A48.
Key words and phrases. Nonnegative Matrices, Companion Matrices, Eigenvalues.
This work was supported by Science Foundation Ireland under Grant SFI RFP 2005
Mat00040.
93