A norm inequality for pairs of commuting positive semidefinite matrices

TL;DR

This paper establishes a new inequality relating the unitarily invariant norm of a sum of products of commuting positive semidefinite matrices to the product of their sums, extending matrix norm inequalities.

Contribution

It introduces a novel norm inequality for sums of products of commuting positive semidefinite matrices, generalizing existing matrix norm bounds.

Findings

Proves the inequality for any unitarily invariant norm.

Shows the inequality holds for matrices with commuting pairs.

Provides a new tool for analyzing matrix norm bounds.

Abstract

For $k=1,\ldots,K$, let $A_k$ and $B_k$ be positive semidefinite matrices such that, for each $k$, $A_k$ commutes with $B_k$. We show that, for any unitarily invariant norm, \[ |||\sum_{k=1}^K A_kB_k||| \le ||| (\sum_{k=1}^K A_k)\;(\sum_{k=1}^K B_k)|||. \]