Operators without eigenvalues in finite-dimensional vector spaces

We introduce the concept of a canonical subspace of C d [ z ] and among other results prove the following statements. An operator in a finite-dimensional vector space has no eigenvalues if and only if it is similar to the operator of multiplication by the independent variable on a canonical subspace of C d [ z ] . An operator in a finite-dimensional Pontryagin space is symmetric and has no eigenvalues if and only if it is isomorphic to the operator of multiplication by the independent variable in a canonical subspace of C d [ z ] with an inner product determined by a full matrix polynomial Nevanlinna kernel.