Characteristic description and numerical computation methods of centrally symmetric matrix algebra
中心对称矩阵代数的特征刻画与数值计算方法

摘要

中心对称矩阵是一类具有特殊结构的矩阵，其在数值分析、图像处理、物理模拟及工程计算等领域具有广泛应用。由于其对称结构在代数计算中展现出独特的规律性，研究其代数特征不仅有助于深入理解矩阵理论，还能显著优化数值计算过程。本文从中心对称矩阵的定义与基本性质出发，深入分析其代数特征与谱结构，并系统探讨相关的数值计算方法，包括特征值求解、线性方程组求解及算法复杂度优化。通过构建结构化分解策略与高效迭代算法，本文旨在为工程计算中涉及中心对称矩阵问题提供理论支持与可行路径。

Abstract

Centrosymmetric matrices are a class of specially structured matrices that find wide applications in numerical analysis, image processing, physical simulation, and engineering computation. Due to their symmetric configuration, these matrices exhibit distinctive patterns in algebraic operations. Studying their algebraic characteristics not only deepens the theoretical understanding of matrix theory but also significantly improves computational efficiency. This paper starts with the definition and basic properties of centrosymmetric matrices, then delves into their algebraic features and spectral structure. It systematically explores numerical computation methods related to such matrices, including eigenvalue computation, linear system solving, and algorithm complexity optimization. By developing structured decomposition strategies and efficient iterative algorithms, the paper aims to provide theoretical support and practical solutions for engineering problems involving centrosymmetric matrices.