Realizations of matrices of rational numbers
type: Item
instance of: scholarly article
title: Realizations of matrices of rational numbers
zbMATH Open document ID: 0606.15006
DOI: 10.1016/0022-314X(87)90023-0
author: Harald K. Wimmer
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: Let W be an \(s\times t\) matrix over the field \({\mathbb{Q}}\) of rational numbers. The author is interested in studying decompositions of W of the form \(W=C(P-N)^{-1} B+D\) where the matrices on the right hand side are integer matrices of dimensions \(s\times m\), \(m\times m\), \(m\times t\) and \(s\times t\), respectively, and P-N is a block diagonal matrix whose i-th block has the form \(p_ iI-N_ i\) with \(p_ i\) prime and \(N_ i\) nilpotent with all entries lying between 0 and \(p_ i-1\). In analogy with the corresponding decomposition in mathematical systems theory over the field F(z) of rational functions, the author calls such a decompositon a state group realization of W. It is easily shown that every W has such a realization. The main results of the paper determine when such a realization is minimal (with respect to m), the relation between such minimal realizations, and how they can be calculated.
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 93C05
zbMATH DE Number: 3979265
zbMATH Keywords: decomposition
zbMATH Keywords: systems theory
zbMATH Keywords: state group realization
zbMATH Keywords: minimal realizations
MaRDI profile type: Publication
cites work: Q3672171
cites work: Q5619754
cites work: Matrices of rational functions
cites work: Dichotomies in stability theory
cites work: Q3905961
cites work: Q3856844
cites work: Q5582663
cites work: Q4771385
cites work: Irreducible Realizations and the Degree of a Rational Matrix
cites work: Q5638871
description: scientific article; zbMATH DE number 3979265