Bitangential interpolation problems for symmetric rational matrix functions
The purpose of this paper is to solve the bitangential (or bidirectional) matrix interpolation problem for rational matrix functions \(F(z)\) with additional types of constraints imposed on \(F\), such as \(F(z)=F(z)^T\), \(F(z)=-F(z)^T\) or \(F(z)^TF(z)=I\). The authors first develop the zero-pole structure of rational matrix functions \(\Theta(s)\) that satisfy an identity of the form \(\Theta(s)^TV\Theta(s)=V\) for some square matrix \(V\) for which \(V=\alpha V^T\), where \(\alpha=\pm 1\). The main tool used is a certain quintuple of matrices \(\tau=(C_\pi,A_\pi;A_\zeta,B_\zeta;\Gamma)\) called a Sylvester data set arising as the null-pole triple for a certain rational matrix function \(\Theta\) which is regular (i.e., \(\Theta\) is square with determinant not identically equal to zero). The matrix function \(\Theta\) is the so-called resolvent matrix for the interpolation problem. For a subset \(\sigma\) of the complex plane, the \(\sigma\)-null-pole triple \(\tau\) of the regular rational matrix \(\Theta\) provides an explicit parametrization (in terms of five matrices) for the module \(S_\sigma(\Theta)\) of rational vector functions given by \(S_\sigma(\Theta)=\Theta{\mathfrak R}^m(\sigma)\), where \({\mathfrak R}^m(\sigma)\) is the space of rational functions that are analytic on \(\sigma\). A calculus of Sylvester data sets and null-pole triples designed to handle a variety of factorization and interpolation problems for rational (or more generally meromorphic) matrix functions is presented. The global structure of the problem is emphasized, and closed formulas for solutions in terms of the original structure of the problem are obtained. As an application, the set of symmetric or anti-symmetric solutions of a set of bitangential interpolation conditions are described. Solutions of bitangential interpolation conditions satisfying other types of symmetries are also discussed.
scientific article; zbMATH DE number 931884
scientific article