Explicit solutions of the discrete-time Lyapunov matrix equation and Kalman-Yakubovich equations

A general solution for the nonsquare nonsymmetric Lyapunov matrix equation in a canonical form is presented. The solution is shown to be a Toeplitz matrix which may be calculated using the backwards Levinson algorithm This solution is then applied to the Kalman-Yakubovich equations to derive a method for generating strictly positive-real functions via the positive-real lemma. This latter result has an application in system identification.