A note on simultaneous block diagonally stable matrices

Abstract

Consider a set of square real matrices of the same size A={A₁, A₂,...,A_N}, where each matrix is partitioned, in the same way, into blocks such that the diagonal ones are square matrices. Under the assumption that the diagonal blocks in the same position have a common Lyapunov solution, sufficient conditions for the existence of a common Lyapunov solution with block diagonal structure for A are presented. Furthermore, as a by-product, an algorithm for the construction of such a common Lyapunov solution is proposed.