Diagonal realizability in the Nonnegative Inverse Eigenvalue Problem

TL;DR

This paper proves that any nonzero spectrum of a diagonalizable nonnegative matrix can be realized by a larger diagonalizable nonnegative matrix of a specific size, advancing understanding of the Nonnegative Inverse Eigenvalue Problem.

Contribution

It establishes a new construction method showing that nonzero spectra can be realized by larger matrices with a specific size, contributing to the diagonal realizability aspect of the NIEP.

Findings

Any nonzero spectrum of a diagonalizable nonnegative matrix can be realized by a larger diagonalizable nonnegative matrix.

The size of the constructed matrix is explicitly given as $k + k^2$ for a spectrum of size $k$.

This result provides a new approach to the diagonal realizability in the Nonnegative Inverse Eigenvalue Problem.

Abstract

We show that if a list of nonzero complex numbers $\sigma=(\lambda_1,\lambda_2,\ldots,\lambda_k)$ is the nonzero spectrum of a diagonalizable nonnegative matrix, then $\sigma$ is the nonzero spectrum of a diagonalizable nonnegative matrix of order $k+k^2.$