On Fillmore's theorem extended by Borobia

Abstract

Fillmore Theorem says that if A is an nxn complex non-scalar matrix and γ_1,...,γ_{n} are complex numbers with γ_1+...+γ_{n}=trA, then there exists a matrix B similar to A with diagonal entries γ_1,...,γ_{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 γ_1,...,γ_{n}. Moreover, we consider the nonnegativity hypothesis and we show that for a list Λ={λ_1,...,λ_{n}} of complex numbers of Suleimanova or Šmigoc type, and a given list Γ={γ_1,...,γ_{n}} of nonnegative real numbers, the remarkably simple condition γ_1+...+γ_{n}=λ_1+...+λ_{n} is necessary and sufficient for the existence of a nonnegative matrix with spectrum Λ and diagonal entries Γ. This surprising simple result improves a condition recently given by Ellard and Šmigoc in arXiv:.1702.02650v1.