The discrete Lyapunov equation in controllable canonical form
Vlastimil Pták · IEEE Transactions on Automatic Control · 1981
A new method of treating Lyapunov equations is proposed. IfDandD^{-}are two self-adjoint one-dimensional matrices related in a certain way, thenX-C^{\ast}XC=Dif and only ifX^{-1}-CX^{-1} C^{\ast}=D^{-}. As an application, a generalization of a recent result is given. Iffis the vectorf=(0,0...,0,1)^{T}, then the solution ofX-CXC^{\ast} = ff^{\ast}is shown to be inverse of the Schur-Cohn matrix.