A sparsity result on nonnegative real matrices with given spectrum

Abstract

Let σ = (λ1, . . . , λn) be the spectrum of a nonnegative real n × n matrix. It is shown that σ is the spectrum of a nonnegative real n× n matrix having at most (n+ 1)2/2− 1 nonzero entries. Let A = (aij) be a real n × n matrix. We say that A is nonnegative if all its entries aij ≥ 0 and that A is positive if all aij > 0. The nonnegative inverse eigenvalue problem (NIEP) is the problem of characterizing those lists σ = (λ1, . . . , λn) of complex numbers λi for which there exists a nonnegative matrix A with spectrum σ(A) = σ. If such an A exists we say that the list σ is realizable and we say that A realizes σ. While considerable work has been done on the NIEP, the problem is still far from being solved and in terms of n, only in the cases n = 2 and n = 3 (Johnson, Loewy and London) has the question been completely settled. See for example [1], [5] for references. For a given list σ = (λ1, . . . , λn), one can attempt to realize σ by the companion matrix C(f) of the polynomial f(x) := (x− λ1) · · · (x− λn) := x + p1x + · · ·+ pn. In this case C(f) is nonnegative if and only if pi ≤ 0 for i = 1, 2, . . . , n. However this condition is very restrictive—it implies for example that f(x) has only one positive real root (see also [2] for a related discussion)—and one can improve the prospects of success by seeking to realize σ by a matrix of the form αIn +C where α ≥ 0 and C is a nonnegative companion matrix. There exist realizable sets σ which are not realizable by matrices of this type. (See Reams’ Thesis [6], Chapter 3 for examples with n = 4.) 1991 Mathematics Subject Classification: Primary 15A48. Secondary 15A33, 15A36. The paper is in final form and no version of it will be published elsewhere.