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On Generalizing Trace Minimization Principles, II

Xin Liang, Ren-Cang Li

Abstract

This paper is concerned with establishing a trace minimization principle for two Hermitian matrix pairs. Specifically, we will answer the question: when is $\inf_X\operatorname{tr}(\widehat AX^{\rm H}AX)$ subject to $\widehat BX^{\rm H}BX=I$ (the identity matrix of apt size) finite? Sufficient and necessary conditions are obtained and, when the infimum is finite, an explicit formula for it is presented in terms of the finite eigenvalues of the matrix pairs. Our results extend Fan's trace minimization principle (1949) for a Hermitian matrix, a minimization principle of Kovač-Striko and Veselić (1995) for a Hermitian matrix pair, and most recent ones by the authors and their collaborators for a Hermitian matrix pair and a Hermitian matrix.

On Generalizing Trace Minimization Principles, II

Abstract

This paper is concerned with establishing a trace minimization principle for two Hermitian matrix pairs. Specifically, we will answer the question: when is subject to (the identity matrix of apt size) finite? Sufficient and necessary conditions are obtained and, when the infimum is finite, an explicit formula for it is presented in terms of the finite eigenvalues of the matrix pairs. Our results extend Fan's trace minimization principle (1949) for a Hermitian matrix, a minimization principle of Kovač-Striko and Veselić (1995) for a Hermitian matrix pair, and most recent ones by the authors and their collaborators for a Hermitian matrix pair and a Hermitian matrix.
Paper Structure (4 sections, 1 theorem, 22 equations)

This paper contains 4 sections, 1 theorem, 22 equations.

Table of Contents

  1. Introduction
  2. Notations.
  3. Preliminaries on a positive semidefinite matrix pair
  4. Main result

Key Result

Theorem 3.1

Given four Hermitian matrices $A,B\in \mathbb{C}^{n\times n},\widehat{A},\widehat{B}\in \mathbb{C}^{\widehat{n}\times \widehat{n}}$ where $n\ge \widehat{n}$, suppose that $\widehat{A}\ne 0$, $A\ne \mu B$ for any $\mu\in \mathbb{R}$, and $\widehat{A}\ne\widehat{\mu}\widehat{B}$ for any $\widehat{\mu} i.e., finite, if and only if one of the following two cases occurs: Moreover, in the first case, w

Theorems & Definitions (3)

  • Definition 2.1: lilb:2013kove:1995
  • Definition 3.1
  • Theorem 3.1