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

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.