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Orthogonal matrix polynomials and system theory. (English) Zbl 0649.93015

Linear and nonlinear mathematical control theory, Conv. Torino/Italy 1987, Rend. Semin. Mat., Torino, Fasc. Spec., 68-124 (1987).
[For the entire collection see Zbl 0636.00007.]
The paper presents a unified approach to the study of orthogonal matrix polynomials associated with Hankel and Toeplitz forms. Given a sequence of \(m\times m\) matrices \(G=\{G_ i\}^{\infty}_{i=1}\) with entries in a (commutative) field F, define for matrix polynomials \[ \Psi (z)=\sum^{n}_{j=0}z\quad j\Psi_ j\quad and\quad \Phi (z)=\sum^{m}_{j=0}z\quad j\Phi_ j \] the bilinear (not necessarily symmetric) form \[ [\Phi,\Psi]_ G = [\Psi_ 0 ... \Psi_ n] \left[\begin{matrix} G_ 1 & G_ 2 & ... & G_{n+1} \\ G_ 2 &&& \vdots \\ \vdots &&& \vdots \\ G_{n+1} && ... & G_{2n+1} \end{matrix}\right] \left[\begin{matrix} \Phi_ 0 \\ \vdots \\ \vdots \\ \Phi_ n \end{matrix}\right]. \] The orthogonal monic matrix polynomials associated with this block Hankel form are studied. In particular, recursive algorithms based on the three term recursion formulas are given. The connections with reproducing kernels, inversion of block Hankel matrices, generalized Bezoutians and the partial realization problem are pointed out. A parallel study is made on the orthogonal matrix polynomials associated with block Toeplitz forms. Here the main problem addressed is a recursive way of inverting finite block Toeplitz matrices. The availability of reproducing kernels lead directly to the generalized Levinson algorithm for inversion of these matrices.
Reviewer: L.Rodman

MSC:

93B25 Algebraic methods
15B57 Hermitian, skew-Hermitian, and related matrices
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
15A63 Quadratic and bilinear forms, inner products
65F05 Direct numerical methods for linear systems and matrix inversion
93B15 Realizations from input-output data

Citations:

Zbl 0636.00007