On Fillmore's theorem extended by Borobia

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.