Algorithms for Triangular Decomposition of Block Hankel and Toeplitz Matrices with Application to Factoring Positive Matrix Polynomials

Abstract

Algorithms are given for calculating the block triangular factors $A,\hat A,B = {A^{ - 1}}$ and $\hat B = {\hat A^{ - 1}}$ and the block diagonal factor D in the factorizations $R = AD\hat A$ and $BR\hat B = D$ of block Hankel and Toeplitz matrices R. The algorithms require $O({p^3}{n^2})$ operations when R is an $n \times n$-matrix of $p \times p$-blocks. As an application, an iterative method is described for factoring $p \times p$-matrix valued positive polynomials $R = \sum \nolimits _{i = - m}^m {R_i}{x^i},{R_{ - i}} = {R’_i}$, as $\bar A(x)\bar A’({x^{ - 1}})$, where $\bar A(x)$ is outer.