On Symmetric Nonnegative Matrices with Prescribed Spectrum1 (original) (raw)

On symmetric nonnegative matrices with prescribed spectrum

International Mathematical Forum, 2014

In this paper we give a sufficient condition for the existence and construction of a symmetric nonnegative matrix with prescribed spectrum, and a sufficient conditon for the existence and construction of a 4 × 4 symmetric nonnegative matrix with prescribed spectrum and diagonal entries. This last condition is independent of the sufficient condition given by Fiedler [LAA 9 (1974) 119-142]. We also give some partial answers on an open question of Guo [LAA 266 (1997) 261-270] about symmetric nonnegative matrices.

Construction of nonnegative symmetric matrices with given spectrum

Linear Algebra and its Applications, 2007

Let σ = (λ 1 , . . . , λ n ) be the spectrum of a nonnegative symmetric matrix A with the Perron eigenvalue λ 1 , a diagonal entry c and let τ = (μ 1 , . . . , μ m ) be the spectrum of a nonnegative symmetric matrix B with the Perron eigenvalue μ 1 . We show how to construct a nonnegative symmetric matrix C with the spectrum (λ 1 + max{0, μ 1 − c}, λ 2 , . . . , λ n , μ 2 , . . . , μ m ).

Existence and construction of nonnegative matrices with complex spectrum

Linear Algebra and its Applications, 2003

The following inverse spectrum problem for nonnegative matrices is considered: given a set of complex numbers σ = {λ 1 , λ 2 , . . . , λ n }, find necessary and sufficient conditions for the existence of an n × n nonnegative matrix A with spectrum σ . Our work is motivated by a relevant theoretical result of Guo Wuwen [Linear Algebra Appl. 266 (1997) 261, Theorem 2.1]: there exists a real parameter λ 0 max 2 j n |λ j | such that the problem has a solution if and only if λ 1 λ 0 . In particular, we discuss how to compute λ 0 and the solution matrix A for certain class of matrices. A sufficient condition for the problem to have a solution is also derived.

Realizability by Symmetric Nonnegative Matrices*

Proyecciones (Antofagasta), 2005

Let Λ = {λ 1 , λ 2 ,. .. , λ n } be a set of complex numbers. The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and sufficient conditions in order that Λ may be the spectrum of an entrywise nonnegative n × n matrix. If there exists a nonnegative matrix A with spectrum Λ we say that Λ is realized by A. If the matrix A must be symmetric we have the symmetric nonnegative inverse eigenvalue problem (SNIEP). This paper presents a simple realizability criterion by symmetric nonnegative matrices. The proof is constructive in the sense that one can explicitly construct symmetric nonnegative matrices realizing Λ.

SYMMETRIC NONNEGATIVE REALIZATION OF SPECTRA

A perturbation result, due to R. Rado and presented by H. Perfect in 1955, shows how to modify r eigenvalues of a matrix of order n, r ≤ n, via a perturbation of rank r, without changing any of the n − r remaining eigenvalues. This result extended a previous one, due to Brauer, on perturbations of rank r = 1. Both results have been exploited in connection with the nonnegative inverse eigenvalue problem. In this paper a symmetric version of Rado's extension is given, which allows us to obtain a new, more general, sufficient condition for the existence of symmetric nonnegative matrices with prescribed spectrum.

Fast construction of a symmetric nonnegative matrix with a prescribed spectrum

Computers & Mathematics with Applications, 2001

In this paper, for a prescribed real spectrum, using properties of the circulant matrices and of the symmetric persymmetric matrices, we derive a fast and stable algorithm to construct a symmetric nonnegative matrix which realizes the spectrum. The algorithm is based on the fast Fourier transform.

On nonnegative realization of partitioned spectra

We consider partitioned lists of real numbers Λ = {λ 1 , λ 2 , . . . , λn}, and give efficient and constructive sufficient conditions for the existence of nonnegative and symmetric nonnegative matrices with spectrum Λ. Our results extend the ones given in [R.L. Soto and O. Rojo. Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem. Linear Algebra Appl., 416:844-856, 2006.] and [R.L. Soto, O. Rojo, J. Moro, and A. Borobia. Symmetric nonnegative realization of spectra. Electron. J. Linear Algebra, 16:1 -18, 2007.] for the real and symmetric nonnegative inverse eigenvalue problem. We also consider the complex case and show how to construct an r × r nonnegative matrix with prescribed complex eigenvalues and diagonal entries.