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Structural Properties of Polynomial and Rational Matrices, a survey

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Abstract (2. Language): 
A review of the structural properties of polynomial and rational ma- trices is presented. After the analysis of the finite spectrum of a polyno- mial matrix A(), via the Smith canonical form, we analyze the infinity as an eigenvalue, but also as a pole or zero (via the Smith McMillan canonical form) when considering A() in the set of rational matrices. Then we focus on the structures generated by the columns of A(). Here we review two different approaches: when considering linear com- binations over the rational functions, and when linear combinations are supposed to be over polynomials only. The objective is to compare and contrast the results of these two lines of thought, as well as to under- line the fundamental differences between matrix polynomials in one or several variables. Structure preserving transformations and equivalence relations over polynomial matrices are also reviewed. With this objec- tive in mind, we also give some insights on the eigenvalue structure of multivariable matrix polynomials.
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