On Fillmore's theorem extended by Borobia

TL;DR

This paper presents a new simple method to compute matrices similar to a given matrix with specified diagonal entries, extending Fillmore's theorem, and establishes a necessary and sufficient condition for the existence of nonnegative matrices with prescribed spectrum and diagonal entries.

Contribution

Introduces a straightforward approach to find similar matrices with given diagonals and proves a simple condition for nonnegative matrices with specific spectra and diagonals.

Findings

A new simple method for computing similar matrices with prescribed diagonals.

Necessary and sufficient condition for nonnegative matrices with given spectrum and diagonals.

Improves upon recent conditions by Ellard and Šmigoc.

Abstract

Fillmore Theorem says that if A is an nxn complex non-scalar matrix and {\gamma}_1,...,{\gamma}_{n} are complex numbers with {\gamma}_1+...+{\gamma}_{n}=trA, then there exists a matrix B similar to A with diagonal entries {\gamma}_1,...,{\gamma}_{n}. Borobia simplifies this result and extends it to matrices with integer entries. Fillmore and Borobia do not consider the nonnegativity hypothesis. Here, we introduce a different and very simple way to compute the matrix B similar to A with diagonal {\gamma}_1,...,{\gamma}_{n}. Moreover, we consider the nonnegativity hypothesis and we show that for a list {\Lambda}={{\lambda}_1,...,{\lambda}_{n}} of complex numbers of Suleimanova or \v{S}migoc type, and a given list {\Gamma}={{\gamma}_1,...,{\gamma}_{n}} of nonnegative real numbers, the remarkably simple condition {\gamma}_1+...+{\gamma}_{n}={\lambda}_1+...+{\lambda}_{n} is necessary and sufficient for the existence of a nonnegative matrix with spectrum {\Lambda} and diagonal entries {\Gamma}. This surprising simple result improves a condition recently given by Ellard and \v{S}migoc in arXiv:.1702.02650v1.