Patterns that require distinct singular values

TL;DR

This paper characterizes matrix patterns where all real matrices have distinct singular values, extending Fiedler's work on eigenvalues of symmetric matrices associated with graphs.

Contribution

It generalizes Fiedler's characterization to broader matrix patterns, identifying those that guarantee distinct singular values for all real matrices with the pattern.

Findings

Characterization of matrix patterns with no repeated singular values

Extension of Fiedler's eigenvalue results to singular values

Identification of patterns requiring distinct singular values

Abstract

Patterns of m by n matrices of term-rank m for which every real matrix with the pattern has no multiple singular value are characterized. This generalizes Fiedler's characterization of the paths being the only graphs for which every real symmetric matrix with the given graph has no repeated eigenvalue.