# On a Schur-type Algorithm for Sequences of Complex \({p} \times{q} \)-matrices and its Interrelations with the Canonical Hankel Parametrization

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## Abstract

Building on work started in [12], we further examine the structure of the set \( \mathcal{H}^{\geq}_{q,2n} \)of all Hankel non-negative definite sequences \( {(s_j)}^{2n}_{j=0} \)of complex \( q \times q \)-matrices. We furthermore examine the important subclasses \( {\mathcal{H}^{{\geq},e}_{q,2n}} \) and \( \mathcal{H}^{\geq}_{q,2n} \), consisting of all Hankel non-negative definite and Hankel positive definite extendable sequences, respectively. These sequence-classes appear naturally when discussing matrix versions of the truncated Hamburger moment problem.

In [12] and [15] a canonical Hankel parametrization \( [{(C_k)^n_{k=1}},{(D_k)^n_{k=0}}], \)consisting of two sequences of complex matrices, was associated with every sequence \( {(s_j)}^{2n}_{j=0} \) of complex \( p \times q \)-matrices. There is a bijective correspondence between the sequence and its canonical Hankel parametrization

Chen and Hu [9] constructed a Schur-type algorithm for a special class of holomorphic matrix-valued functions in the upper half-plane so that matrix versions of the truncated Hamburger moment problem might be dealt with in the degenerate and non-degenerate cases, simultaneously. A closer analysis of their algorithm showed that it implicitly contains an interesting procedure for sequences belonging \( {\mathcal{H}^{{\geq},e}_{q,2n}} \)This procedure serves as the focus of our work here, although we have chosen a slightly different and more general setting.

Our approach is based on a suitable extension of the concept of reciprocal sequences, which are used in power series inversions. We will show that, given *n* as a positive integer, this concept rests on a particular method for producing sequences belonging to \( {\mathcal{H}^{{\geq}}_{q,2n}} \), starting from a sequence \( {{(s_j)}^{2n}_{j=0}}\; \in \;{{\mathcal{H}^{{\geq}}_{q,2n}}} \).Using this, we develop a Schur-type algorithm for finite sequences of complex \( p \times q \)-matrices. We show that the Schur-type algorithm preserves specific subclasses of \( {\mathcal{H}^{{\geq}}_{q,2n}} \), for example: \( {\mathcal{H}^{{\geq},e}_{q,2n}} \) \( {\mathcal{H}^{{\geq}}_{q,2n}} \). One of our main results (see Theorem 9.15) expresses that, given a sequence \( {{(s_j)}^{2n}_{j=0}}\; \in \;{{\mathcal{H}^{{\geq},e}_{q,2n}}} \), the Schur-type algorithm produces, exactly, its canonical Hankel parametrization. This leads us to a deeper understanding of the canonical Hankel parametrization.

## Keywords

Non-negative Hermitian measures truncated matricial Hamburger moment problem non-negative Hermitian block Hankel matrices Hankel nonnegative definite sequences Hankel non-negative definite extendable sequences Hankel non-negative definite extendable sequences canonical Hankel parametrization reciprocal sequences Schur-type algorithm.## Preview

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