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Linear preserver problems in matrix positivity theory

Projesh Nath Choudhury, Shivangi Yadav

Abstract

Linear preserver problems have been a central focus of research in matrix theory and operator theory for more than a century, beginning with Frobenius' 1897 characterization of determinant preserving linear maps on the space of complex matrices. Since this foundational result, considerable work has examined linear preservers of diverse subsets, functions, and relations across different matrix and operator spaces. The purpose of this survey is to present the current state of research on linear preserver problems for several positivity classes of matrices. We provide an overview of recent developments in the literature and, for each positivity class considered, identify gaps that remain to guide future research.

Linear preserver problems in matrix positivity theory

Abstract

Linear preserver problems have been a central focus of research in matrix theory and operator theory for more than a century, beginning with Frobenius' 1897 characterization of determinant preserving linear maps on the space of complex matrices. Since this foundational result, considerable work has examined linear preservers of diverse subsets, functions, and relations across different matrix and operator spaces. The purpose of this survey is to present the current state of research on linear preserver problems for several positivity classes of matrices. We provide an overview of recent developments in the literature and, for each positivity class considered, identify gaps that remain to guide future research.
Paper Structure (9 sections, 32 theorems, 44 equations)

This paper contains 9 sections, 32 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Linear maps preserving positivity classes of matrices
  3. Positive semidefinite matrices
  4. Completely positive rank preservers
  5. Copositive matrices
  6. Sign regular matrices
  7. M-matrices, Inverse M-matrices, and H-matrices
  8. P-matrices, $\mathcal{D}$-stable matrices, and $\mathcal{A}$-matrices
  9. Semipositive matrices

Key Result

Lemma 2.2

BHJ85 Let $V$ be a finite-dimensional vector space and let $\mathcal{S}\subseteq V$. Suppose that $P(\mathcal{S})$ denotes the set of all onto $\mathcal{S}$-preservers. Then $P(\mathcal{S})\subseteq P(\overline{\mathcal{S}})$.

Theorems & Definitions (52)

  • Definition 2.1
  • Lemma 2.2
  • Example 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Corollary 2.6
  • proof
  • Definition 2.7
  • Theorem 2.8
  • Theorem 2.9
  • ...and 42 more