Rigorous Perturbation Bounds for the QX Decomposition for Centrosymmetric Matrices

TL;DR

This paper derives explicit, rigorous perturbation bounds for the QX factorization of centrosymmetric matrices, providing insights into stability under perturbations with both norm-wise and component-wise analysis.

Contribution

It introduces explicit perturbation bounds for the QX factorization of centrosymmetric matrices, including weak and strong bounds, and condition numbers, using matrix-equation and fixed-point approaches.

Findings

Derived weak and strong perturbation bounds.

Explicit expressions for condition numbers.

Numerical tests confirm theoretical results.

Abstract

Konrad Burnik suggests a structure-preserving QR factorization for centrosymmetric matrices, known as QX factorization. In this article, we obtain the explicit expressions for rigorous perturbation bounds of the QX factorization when the original matrix is perturbed, either norm-wise or component-wise. First, using the matrix-equation approach, weak rigorous perturbation bounds are derived. Then, strong rigorous perturbation bounds are obtained by combining the modified matrix-vector equation approach with the strategy for the Lyapunov majorant function and the Banach fixed-point theorem. The mixed and component-wise condition numbers and their upper bounds are also explicitly expressed. Numerical tests illustrate the validity of the obtained results.