Realizations of matrices of rational numbers
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.
scientific article; zbMATH DE number 3979265