Perturbation estimation for the parallel sum of Hermitian positive semi-definite matrices

TL;DR

This paper introduces a new upper bound for the perturbation of the parallel sum of Hermitian positive semi-definite matrices, providing sharper bounds and a factorization formula to analyze stability under perturbations.

Contribution

It presents a novel common upper bound for perturbed parallel sums and derives a factorization formula, advancing the understanding of perturbation effects in matrix analysis.

Findings

New common upper bound for perturbed parallel sum

Sharp norm upper bounds for perturbations

Numerical validation of bounds' sharpness

Abstract

Let $\mathbb{C}^{n\times n}$ be the set of all $n \times n$ complex matrices. For any Hermitian positive semi-definite matrices $A$ and $B$ in $\mathbb{C}^{n\times n}$, their new common upper bound less than $A+B-A:B$ is constructed, where $(A+B)^\dag$ denotes the Moore-Penrose inverse of $A+B$, and $A:B=A(A+B)^\dag B$ is the parallel sum of $A$ and $B$. A factorization formula for $(A+X):(B+Y)-A:B-X:Y$ is derived, where $X,Y\in\mathbb{C}^{n\times n}$ are any Hermitian positive semi-definite perturbations of $A$ and $B$, respectively. Based on the derived factorization formula and the constructed common upper bound of $X$ and $Y$, some new and sharp norm upper bounds of $(A+X):(B+Y)-A:B$ are provided. Numerical examples are also provided to illustrate the sharpness of the obtained norm upper bounds.