Minimal symmetric factorizations of symmetric real and complex rational matrix functions Original Research Article

Real and complex rational matrix functions W(λ) such that W(λ) = ξ[W(ηλ)]T, where ξ,η = ± 1, and their minimal factorizations W(λ) = [L(ηλ)]TD(λ)L(λ) are studied. Such factorizations are described geometrically in terms of invariant subspaces with certain isotropic properties with respect to a quadratic form. Using results (partly known and partly proved in the paper) concerning the dimensions of such subspaces, upper bounds are given for the degrees of the rational matrix functions L(λ) in the above factorizations.