On minimal bases and indices of rational matrices and their linearizations
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This paper presents a complete theory of the relationship between the minimal bases and indices of rational matrices and those of their strong linearizations. 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. These first results extend under different assumptions pioneer results obtained by Verghese, Van Dooren and Kailath in 1979-80, which have not been sufficiently recognized in the modern literature in our opinion. Next, 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, since only the total sums of the minimal indices are related to each other in the strong case. In contrast, if the specific families of strong block minimal bases linearizations and M_1 and M_2-strong linearizations are considered, then simple relationships between the minimal bases and indices of the linearizations and the rational matrices are obtained. The relevant influence of the work of Paul Van Dooren and coworkers on these topics is emphasized throughout this paper.
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