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