A Semi-Definite Lyapunov Theorem and the Characterization of Tridiagonal D-Stable Matrices

A new necessary and sufficient condition is given for an $n \times n$ complex matrix A to be stable. It involves a positive semi-definite image under a Lyapunov map and the real and imaginary parts of A. This condition is then used to characterize the real tridiagonal matrices which are D-stable, and those which are totally D-stable.