On common diagonal Lyapunov solutions

Several recent results regarding common diagonal Lyapunov solutions are further explored here. The first one, attributed to Redheffer and revisited by Shorten and Narendra, reduces the diagonal stability of a matrix to common diagonal Lyapunov solutions on two matrices of order one less. We present a shorter, purely matrix-theoretic proof of this result along with its extensions. The second one concerns two different necessary and sufficient conditions, due to Oleng, Narendra, and Shorten, for a pair of 2×22×2 matrices to share a common diagonal Lyapunov solution. We show that these two conditions are connected directly to each other.