Root vectors of polynomial and rational matrices: theory and computation (Q2093531)

Consider the field of rational functions \(F(\lambda )\) over an algebraically closed field \(F\). In this context, a rational matrix is a matrix with entries in \(F(\lambda )\). Let \(R(\lambda )\) be a nonzero rational \(m\times n\) matrix and \(\lambda _{0}\in F\). Localizing \(F(\lambda )\) at \(\lambda -\lambda _{0}\) we obtain the local subring \(\mathcal{R}:=\{a(\lambda )/b(\lambda )\mid a(\lambda ),b(\lambda )\in F[\lambda ]\) and \(b(\lambda _{0})\neq 0\}\). Then it is known that there exist invertible \(m\times m\) and \(n\times n\) matrices \(A(\lambda ) \) and \(B(\lambda )\) over \(\mathcal{R}\) such that \(A(\lambda )R(\lambda )B(\lambda )\) has the form \[ A(\lambda )R(\lambda )B(\lambda )=\left( \begin{array}{cc} D & 0 \\ 0 & 0 \end{array} \right), \] where \(D\) is a square diagonal matrix whose diagonal entries all have the form \((\lambda -\lambda _{0})^{\sigma }\) with \(\sigma \in \mathbb{Z}\) (the local Smith-McMillan form of \(R(\lambda )\) at \(\lambda _{0}\)). The exponents \(\sigma \) appearing in \(D\) are uniquely determined by \(R(\lambda )\) and are called the structural indices. This paper considers the problem of determining this decomposition of \(R(\lambda )\) at \(\lambda _{0}\). A recent paper by \textit{F. M. Dopico} and \textit{V. Noferini} [Linear Algebra Appl. 584, 37--78 (2020; Zbl 1439.15004)] uses the concept of root polynomials to study this problem in the case where the entries of \(R(\lambda )\) lie in \(F[\lambda ]\). In the present paper, the authors show that most of the results for polynomial matrices extend naturally to rational matrices \(R(\lambda )\) even when \(R(\lambda )\) is singular or has coalescent poles and zeros, and to infinity as well as the finite points \(\lambda _{0}\). A key tool is construction of a minimal state space realizing \(R(\lambda )\). The results are illustrated with computations in the case \(F=\mathbb{C}.\)