### abstract

- We here specialize the standard matrix-valued polynomial interpolation to the case where on the imaginary axis the interpolating polynomials admit various symmetries: Positive semidefinite, Skew-Hermitian, $ J $-Hermitian, Hamiltonian and others. The procedure is comprized of three stages, illustrated through the case where on $ i\R $ the interpolating polynomials are to be positive semidefinite. We first, on the expense of doubling the degree, obtain a minimal degree interpolating polynomial $ P (s) $ which on $ i\R $ is Hermitian. Then we find all polynomials $\Psi (s) $, vanishing at the interpolation points which are positive semidefinite on $ i\R $. Finally, using the fact that the set of positive semidefinite matrices is a convex subcone of Hermitian matrices, one can compute the minimal scalar $\hat {\beta}\geq 0$ so that $ P (s)+\beta\Psi (s) $ satisfies all interpolation constraints for …