On maximal weight solutions of two matricial moment problems in the nondegenerate case

We first revisit the question of existence and uniqueness of maximal weight solutions of the truncated trigonometric matrix moment problem (Problem (TTM)) in the nondegenerate case. The starting point of the consideration is a specialized representation of the matrix weight assigned at each point on the unit circle for the solutions of (nondegenerate) Problem (TTM). Our approach is essentially self-contained and different from the existing methods. Moreover, based on this approach we also derive some important weight features of the so-called N-extremal solutions within the whole solution set of (nondegenerate) Problem (TTM) and the discrete structure of them. In the second part, these observations inspire us to look for the corresponding weight features of the solutions of a (nondegenerate) moment problem for rational matrix-valued functions (Problem (R)), where only the theory of orthogonal matrix polynomials on the unit circle is involved.