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Schur-Horn theorem and Ky Fan's minimum principle for symplectic eigenvalues

Kennett L. Dela Rosa, Aedan Jarrod A. Potot

Abstract

The symplectic analogues of Schur's theorem and Ky Fan's minimum principle are shown to be equivalent. Moreover, the symplectic Schur's and Horn's theorems are extended to generalized means.

Schur-Horn theorem and Ky Fan's minimum principle for symplectic eigenvalues

Abstract

The symplectic analogues of Schur's theorem and Ky Fan's minimum principle are shown to be equivalent. Moreover, the symplectic Schur's and Horn's theorems are extended to generalized means.
Paper Structure (5 sections, 10 theorems, 17 equations)

This paper contains 5 sections, 10 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symplectic eigenvalues
  4. Symplectic Schur-Horn theorem
  5. Symplectic Ky Fan's minimum principle

Key Result

Proposition 2.4

Let $U\in M_{2m}$ and $V,W\in M_{2n}$ be symplectic. Then the following are also symplectic: $W^\top$, $W^{-1}$, $VW$, and $U\boxplus V$.

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Definition 2.7
  • ...and 14 more