Canonical matrix factorisation and polynomial Riccati equations
Let \(R(z)\) be a symmetric \(n\times n\) polynomial matrix, and let \(\mu= (M_1, \dots, M_n)\), \(M_k\) being positive integers. A \(\mu\)-factorization of \(R(z)\) means writing \(R(z)= \Pi^*(z) C\Pi(z)\), where the \(k\)th column of \(\Pi(z)\) has degree at most \(M_k\). The main results state that: a) The successive factor extraction algorithm adapted to \(\mu\)-factorization does always find a solution if it exists. b) Existence of a \(\mu\)-factorization is equivalent to the existence of a solution to a corresponding polynomial Riccati equation. c) For any minimal state-space realization the existence of a \(\mu\)-factorization is equivalent to the existence of an antistabilising solution of the adjoined Riccati equation. d) A criterion for existence of a \(\mu\)-factorizaion is expressed in terms of poles of the matrix \(R(z)^{-1}\).
scientific article; zbMATH DE number 1057557
scientific article