Canonical forms and strictly positive real functions for discrete multivariable systems

In this paper, we show that, for the discrete multivariable systems, the solution of the Lyapunov equation in the controllable canonical form transforms a companion matrix to another companion matrix. We also show that the first (last)
m
columns of the inverse of this solution matrix, where the block size is
m \times m
, give the left (right) matrix fraction decomposition of a strictly positive real function.