Finite and infinite structures of rational matrices: a local approach

Abstract

The structure of a rational matrix is given by its Smith-McMillan invariants. Some properties of the Smith-McMillan invariants of rational matrices with elements in different principal ideal domains are presented: In the ring of polynomials in one indeterminate (global structure), in the local ring at an irreducible polynomial (local structure), and in the ring of proper rational functions (infinite structure). Furthermore, the change of the finite (global and local) and infinite structures is studied when performing a Mobius transformation on a rational matrix. The results are applied to define an equivalence relation in the set of polynomial matrices, with no restriction on size, for which a complete system of invariants are the finite and infinite elementary divisors.

Möbius transformations of matrix polynomials

“…This relationship can be conveniently and concisely expressed by comparing the Jordan characteristics and Smith forms of P and M A (P ); these comparisons constitute the two main results of this section, Theorems 5.3 and 5.7. Alternative approaches to this elementary divisor question are taken in [5], [6], and [66], with results analogous to those in this section. In [66], matrix polynomials and their Smith forms are treated in homogeneous form, but the only choice for grade considered is grade P = deg P .…”

“…In [66], matrix polynomials and their Smith forms are treated in homogeneous form, but the only choice for grade considered is grade P = deg P . The analysis is extended in [5] and [6] to include not just matrices with entries in a ring of polynomials, but to rational matrices, and even matrices with entries from an arbitrary local ring. See also Remark 5.5 for an extension to more general rational transforms.…”

“…Our results generalize and unify recent and classical results on how Möbius transformations change the finite and infinite elementary divisors of matrix pencils and matrix polynomials [5,15,65,66]; see also [53] for an extension to more general rational transformations. We note that Möbius transformations are also used to study proper rational matrix-valued functions and the SmithMcMillan form [4,6,24,64], but we do not discuss this topic here.…”

Linearizations of matrix polynomials in Newton bases

“…3.2], are replaced in Theorem 2.12 by the algebraically defined class of q -prime matrices, where q(λ) = λ − λ 0 . On the other hand, a more general version of Theorem 2.12 that holds for rational matrices is given in [2]. In the language of [2], for a fixed non-constant F-irreducible scalar polynomial q(λ), the q -prime matrices from Definition 2.9 are elements of GL n (F q (λ)).…”

“…In the language of [2], for a fixed non-constant F-irreducible scalar polynomial q(λ), the q -prime matrices from Definition 2.9 are elements of GL n (F q (λ)). With this understanding, Theorem 2.12 can then be seen to be a special case of [2,Thm. 4.2].…”