2018/04/16 by Ana I. Julio, Julio, Ana I., Ricardo L. Soto +1 · 1 citation
Computer Science · Mathematics · #Algebraic and Geometric Analysis #FOS: Mathematics #Mathematical economics #Mathematics #Matrix Theory and Algorithms #Pure mathematics #Spectral Theory (math.SP) #advanced mathematical theories #math.SP
paper · pdf · doi:10.48550/arxiv.1804.05738
published in arXiv (Cornell University) (Cornell University)
arxiv created 2018/04/16 · openalex publication_date 2018/04/16 · arxiv updated 2018/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fillmore Theorem says that if A is an nxn complex non-scalar matrix and γ1,...,γn are complex numbers with γ1+...+γn=trA, then there exists a matrix B similar to A with diagonal entries γ1,...,γn. 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 γ1,...,γn. Moreover, we consider the nonnegativity hypothesis and we show that for a list Λ=λ1,...,λn of complex numbers of Suleimanova or Šmigoc type, and a given list Γ=γ1,...,γn of nonnegative real numbers, the remarkably simple condition γ1+...+γn=λ1+...+λn 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.