正定Hermite矩阵流形上代数Lyapunov方程的信息几何算法

An Information Geometric Algorithm for Algebraic Lyapunov Equation on Positive-Definite Hermitian Matrix Manifold

摘要: 对于正定Hermite矩阵流形上的代数Lyapunov方程 A ^H X + XA + P =0, 基于流形的黎曼几何结构, 作者以矩阵- A ^H X + XA 和 P 之间的测地距离为目标函数, 提出了代数Lyapunov方程数值解的信息几何算法. 最后,给出了正定Hermite矩阵流形上的代数Lyapunov方程的数值模拟结果.

Abstract: For the algebraic Lyapunov equation A ^H X + XA + P =0 on the positive-definite Hermitian matrix manifold, the information geometric algorithm for the numerical solution of the algebraic Lyapunov equation was given via the Riemannian geometric structure of the manifold. In this algorithm, the geodesic distance between the matrix - A ^H X + XA and P was adopted as the cost function. Finally, the information geometric algorithm was illustrated for the algebraic Lyapunov equation on the positive-definite Hermitian matrix manifold by some numerical simulation examples.