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Patterns that require distinct singular values

Caleb Cheung, Bryan Shader

TL;DR

The paper investigates when zero-nonzero patterns of real matrices force all singular values to be simple, extending Fiedler's eigenvalue results to the singular-value setting via patterns. It introduces the Strong Singular Value Property (SSVP), the Matrix Liberation Theorem, and direct-sum techniques to transfer singular-value properties to superpatterns, enabling pattern-based classifications. It fully characterizes square patterns with full term-rank that force simple singular values: such patterns correspond to Fiedler-graph bigraphs, with a precise weak-path condition providing a tight structural criterion. The rectangular (m by n) extension shows the same Fiedler-graph mechanism suffices for simplicity of singular values under full term-rank, supported by block-decomposition and matching arguments. Finally, it initiates study of square patterns with term-rank $m-1$, giving partial results for avoiding zero singular-value multiplicity and constructing families that admit multiple nonzero singular values, exemplified by a Hessenberg-type orthogonal pattern $H_n$.

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.

Patterns that require distinct singular values

TL;DR

The paper investigates when zero-nonzero patterns of real matrices force all singular values to be simple, extending Fiedler's eigenvalue results to the singular-value setting via patterns. It introduces the Strong Singular Value Property (SSVP), the Matrix Liberation Theorem, and direct-sum techniques to transfer singular-value properties to superpatterns, enabling pattern-based classifications. It fully characterizes square patterns with full term-rank that force simple singular values: such patterns correspond to Fiedler-graph bigraphs, with a precise weak-path condition providing a tight structural criterion. The rectangular (m by n) extension shows the same Fiedler-graph mechanism suffices for simplicity of singular values under full term-rank, supported by block-decomposition and matching arguments. Finally, it initiates study of square patterns with term-rank , giving partial results for avoiding zero singular-value multiplicity and constructing families that admit multiple nonzero singular values, exemplified by a Hessenberg-type orthogonal pattern .

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.

Paper Structure

This paper contains 4 sections, 21 theorems, 52 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Square Patterns Requiring Simple Singular Values
  3. Non-square patterns requiring all simple singular values
  4. Square Patterns with less than Full Term-Rank

Key Result

Theorem 2.1

Let $A$ have the SSVP and $P$ be a superpattern of the pattern of $A$. Then there exists a matrix with pattern $P$ and singular values $\Sigma(A)$.

Figures (7)

  • Figure 1: Digraph of pattern in \ref{['inout']}
  • Figure 2: The Paw Graph
  • Figure 3: Decomposition into directed paths
  • Figure 4: Graph of $PP^\top$
  • Figure 5: The subdivided claw
  • ...and 2 more figures

Theorems & Definitions (43)

  • Theorem 2.1: Superpattern Theorem
  • Theorem 2.2: Matrix Liberation Theorem
  • Theorem 2.3: Direct Sum Theorem
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Corollary 2.6
  • proof
  • Lemma 2.7
  • ...and 33 more