Two results about the matrix exponential (Q1361832)

The author characterizes the matrices \(A\) whose exponential is normal. It turns out that the matrix \(A\) itself need not be normal. Also an upper bound of \(\text{trace} (e^Ae^{A^*})\) is obtained for a stable matrix \(A\), in terms of a corresponding Lyapunov equation; the method provides bounds that are asymptotically stable.