The discrete Lyapunov equation in controllable canonical form

A new method of treating Lyapunov equations is proposed. If
D
and
D^{-}
are two self-adjoint one-dimensional matrices related in a certain way, then
X-C^{\ast}XC=D
if and only if
X^{-1}-CX^{-1} C^{\ast}=D^{-}
. As an application, a generalization of a recent result is given. If
f
is the vector
f=(0,0...,0,1)^{T}
, then the solution of
X-CXC^{\ast} = ff^{\ast}
is shown to be inverse of the Schur-Cohn matrix.