A new characterization of simultaneous Lyapunov diagonal stability via Hadamard products

A well-known characterization by Kraaijevanger [14] for Lyapunov diagonal stability states that a real, square matrix A is Lyapunov diagonally stable if and only if A∘S is a P-matrix for any positive semidefinite S with nonzero diagonal entries. This result is extended here to a new characterization involving similar Hadamard multiplications for simultaneous Lyapunov diagonal stability on a set of matrices. Among the main ingredients for this extension are a new concept called P-sets and a recent result regarding simultaneous Lyapunov diagonal stability by Berman, Goldberg, and Shorten [2].