J-unitary preserving automorphisms of rational matrix functions: state space theory, interpolation, and factorization

Abstract

Let GL m ( R ) be the multiplicative group of C m× m valued rational functions of one complex variable z with determinant not vanishing identically. We analyze general inner automorphisms of GL m ( R ) that preserve the set of matrix functions whose values are unitary (with respect to an indefinite scalar product) for real, or unimodular, values of z. The analysis is based on state space representations. In particular, homogeneous interpolation problems are studied for the rational matrix functions associated with such a general inner automorphism. We also develop results on minimal factorizations within the group associated with J-unitary preserving generalized inner automorphism.