Weak supermajorization between symplectic spectra of positive definite matrix and its pinching

TL;DR

This paper establishes weak supermajorization relations between symplectic spectra of positive definite matrices and their block structures, revealing new spectral inequalities in matrix analysis.

Contribution

It introduces novel weak supermajorization relations involving symplectic eigenvalues and block matrices, expanding understanding of spectral inequalities.

Findings

Proves that $d(E igoplus G) ext{ is weakly supermajorized by } d(A)$.

Establishes a weak supermajorization relation between eigenvalues of certain matrix functions.

Provides insights into spectral inequalities for positive definite matrices with block structures.

Abstract

Let $A = \begin{bmatrix} E & F \\ F^T & G \end{bmatrix}$ be a $2n \times 2n$ real positive definite matrix, where $E, F,$ and $G$ are $n \times n$ blocks. It is shown that $\ d(E \oplus G) \prec^w d(A)$. Here $d(A)$ denotes the $n$-vector consisting of the symplectic eigenvalues of $A$ arranged in the non-decreasing order. We also observe the following weak supermajorization relation, which is interesting on its own: $ \lambda \left( \left(\mathscr{C}(G)^{1/2} \mathscr{C}(E) \mathscr{C}(G)^{1/2}\right)^{1/2} \right) \prec^w \lambda \left( \left(G^{1/2} E G^{1/2} \right)^{1/2} \right)$. Here $\lambda \left( \left( G^{1/2}E G^{1/2} \right)^{1/2} \right)$ denotes the $n$-vector with entries given by the eigenvalues of $\left( G^{1/2}E G^{1/2} \right)^{1/2}$ in the non-decreasing order.