External problems for Ho¨lder norms of matrices and realizations of linear systems

Let F and G be complex $n \times n$ matrices and $\nu ( \cdot )$ be a matrix norm. We consider the functional $\mu ( F,G;S ) = \nu ( FS )\nu ( S^{ - 1} G )$, where S varies over all nonsingular $n \times n$ matrices. For certain singular value norms $\nu$ the infimum of $\mu $ is determined. An application to realizations of linear systems is given.