The real and the symmetric nonnegative inverse eigenvalue problems are different

Abstract

Abstract. We show that there exist real numbers λ 1 , λ 2 , . . . , λn that occur as the eigenvalues of an entry-wise nonnegative n-by-n matrix but do not occur as the eigenvalues of a symmetric nonnegative n-by-n matrix. This solves a problem posed by Boyle and Handelman, Hershkowitz, and others. In the process, recent work by Boyle and Handelman that solves the nonnegative inverse eigenvalue problem by appending 0's to given spectral data is refined. 1.Let M m,n (M m,n (R)) denote the set of all m-by-n complex (real) matrices, and letThe nonnegative inverse eigenvalue problem (NIEP) asks which sets of n complex numbers λ 1 , λ 2 , . . . , λ n occur as the eigenvalues (spectrum) of some entry-wise nonnegative matrix A ∈ M n . In case the data: λ 1 , . . . , λ n are real, two natural variations suggest themselves:(1) The real nonnegative inverse eigenvalue problem (RNIEP) asks which sets of n real numbers occur as the spectrum of a nonnegative A ∈ M n ; and (2) the symmetric nonnegative inverse eigenvalue problem (SNIEP) asks which sets of n real numbers occur as the spectrum of a symmetric nonnegative matrix A ∈ M n .Each problem remains open in general. Several necessary conditions for NIEP are known. Suppose that λ 1 , . . . , λ n are complex numbers. For every positive integer k define the momentsIf λ 1 , . . . , λ n are the eigenvalues of an n-by-n nonnegative (positive) matrix A, we must have S k (λ) ≥ 0 (S k (λ) > 0) for every positive integer k, because S k (λ) is just the trace of A k . The Perron-Frobenius theorem implies that max 1≤i≤n |λ i | is an eigenvalue of A. (The Perron-Frobenius condition and the nonnegativity of the moments are not independent; see a remark after the statement of Theorem 2.) We also have:

The role of certain Brauer and Rado results in the nonnegative inverse spectral problems

“…In this section we consider the symmetric nonnegative inverse eigenvalue problem (SNIEP). It is well known that the RNIEP and the SNIEP are equivalent for n ≤ 4, while they are different for n ≥ 5 [12]. The first results about symmetric nonnegative realization are due to Fiedler [10].…”

“…If the realizing matrix is required to be symmetric we have the symmetric nonnegative inverse eigenvalue problem (SNIEP), which has been solved for n = 5 with realizing matrices of trace zero by Spector [43]. For n ≤ 4, the RNIEP and the SNIEP are equivalent, while for n ≥ 5, they are different [12]. A number of sufficient conditions for the existence of a symmetric nonnegative matrix with prescribed spectrum have also been obtained (see [33,34,38] and the references therein).…”

On Open Problems of Nonnegative Inverse Eigenvalues Problem

“…pdf). Readers also may refer to [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] for some previous results. In some articles, some necessary conditions and sufficient conditions for the three problems above have been given under some small dimension or special cases [7].…”

A characterization of trace zero bisymmetric nonnegative $5 \times 5$ matrices

“…In [16], Wuwen showed that the RNIEP and the SNIEP are equivalent when n ≤ 4. The fact that the RNIEP and the SNIEP are different for n > 4 was proved by Johnson et al in [3]. The NIEP for 4 × 4 traceless matrices was solved by Reams in [10] and 5 × 5 traceless matrices was solved by Laffey and Meehan in [6].…”