On semiscalar equivalence of lower order polynomial matrices


Bogdan Shavarovskii

Department of Algebra, Pidstryhach
Institute for Applied Problems of
Mechanics and Mathematics of
NAS of Ukraine


November 29, 2016 at 15:05 in Lecture Room 377

Abstract of talk

Let \mathbb{C} be the field of complex numbers and A(x),B(x) be n \times n matrices with the entries in the polynomial ring \mathbb{C}[x]. Then A(x) is said to be semiscalarly equivalent to B(x) if there exist matrices P in \mathrm{GL}(n,\mathbb{C}) and Q(x) in \mathrm{GL}(n,\mathbb{C}[x]) (i.e. \mathrm{det}Q(x) is a nonzero complex number) such that A(x)=PB(x)Q(x). The semiscalarity concept is itself interesting as it occurs naturally in the various problems in the applied mathematics and engineering. However, the problem of finding a complete set of (computable) semiscalarity equivalence invariants is very difficult. In this report we are going to discuss some classes of 2 \times 2 and 3 \times 3 polynomial matrices such that it is possible to obtain the complete invariants system for their elements, and the canonical forms for such matrices with respect to semiscalar equivalence are indicated as well.

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