The Diagonalizable Nonnegative Inverse Eigenvalue Problem

Abstract

Abstract:In this article we provide some lists of real numbers which can be realized as the spectra of nonnegative diagonalizable matrices but which are not the spectra of nonnegative symmetric matrices. In particular, we examine the classical list σ = ( + t, − t, − , − , − ) with t ≥ , and show that σ is realizable by a nonnegative diagonalizable matrix only for t ≥ . We also provide examples of lists which are realizable as the spectra of nonnegative matrices, but not as the spectra of nonnegative diagonalizable matrices by examining the Jordan Normal Form.

On universal realizability of spectra

“…The construction of a counterexample is based on the study of UR lists of size 5 with trace zero and three negative elements. This construction has been motivated by the work of Cronin and Laffey [4]. They show that a realizable list is not necessarily diagonalizably realizable.…”

“…In [4], Cronin and Laffey examine the subtle difference between the symmetric nonnegative inverse eigenvalue problem (SNIEP), in which the realizing matrix is required to be symmetric, and the real diagonalizable nonnegative inverse eigenvalue problem (DRNIEP), in which the realizing matrix is diagonalizable. The authors in [4] give examples of lists of real numbers, which can be the spectrum of a nonnegative matrix, but not the spectrum of a diagonalizable nonnegative matrix.…”

Permutative universal realizability

“…. , λn} of complex numbers, is said to be diagonalizably realizable (DR), if there is a diagonalizable realizing matrix for Λ [2]. Moreover, Λ is said to be universally realizable (UR), if it is realizable for each possible Jordan canonical form (JCF) allowed by Λ.…”

On real algebras generated by positive and nonnegative matrices

“…This is the widely known nonnegative inverse eigenvalue problem which is of particular importance. See [7,12] for reviews of the problem and works [1,4,5,8] for some recent results. The present paper treats a generalization of the aforementioned problems.…”