Tangent spaces of rational matrix functions

We consider the task of determining tangent spaces for classes of rational matrix valued functions. Our analysis is based on methods from control theory, and in particular the theory of polynomial models. Explicit descriptions of tangent spaces of rational transfer functions, stable rational transfer functions, rational inner functions, and symmetric rational transfer functions are obtained. Moreover, a new proof of Delchampsʹs decomposition formula for the tangent bundle of rational transfer functions is given. A Riemannian metric as well as a symplectic structure is defined.