On minimal bases and indices of rational matrices and their linearizations

TL;DR

This paper develops a comprehensive theory linking the minimal bases and indices of rational matrices with their strong linearizations, enabling effective computation of these properties through specific linearization families.

Contribution

It establishes new relationships between rational matrices and their linearizations, especially for strong block minimal bases linearizations, extending previous results to rectangular matrices.

Findings

Relationships between minimal bases and indices of rational matrices and their linearizations are established.

Strong block minimal bases linearizations allow for efficient computation of minimal bases and indices.

Results extend to rectangular rational matrices, broadening applicability.

Abstract

A complete theory of the relationship between the minimal bases and indices of rational matrices and those of their strong linearizations is presented. Such theory is based on establishing first the relationships between the minimal bases and indices of rational matrices and those of their polynomial system matrices under the classical minimality condition and certain additional conditions of properness. This is related to pioneer results obtained by Verghese, Van Dooren and Kailath in 1979-80, which were the first proving results of this type under different nonequivalent conditions. It is shown that the definitions of linearizations and strong linearizations do not guarantee any relationship between the minimal bases and indices of the linearizations and the rational matrices in general. In contrast, simple relationships are obtained for the family of strong block minimal bases linearizations, which can be used to compute minimal bases and indices of any rational matrix, including rectangular ones, via algorithms for pencils. These results extend the corresponding ones for other families of linearizations available in recent literature for square rational matrices.