Matricial Gaussian quadrature rules: nonsingular case

Let $L$ be a linear operator on univariate polynomials of bounded degree, mapping into real symmetric matrices, such that its moment matrix is positive definite. It is known that $L$ admits a finitely atomic positive matrix-valued representing measure $\mu$. Any $\mu$ with the smallest sum of the ranks of the matricial masses is called minimal. In this paper, we characterize the existence of a minimal representing measure containing a prescribed atom with prescribed rank of the corresponding mass, thus extending a recent result (2020) for the scalar-valued case. As a corollary, we obtain a constructive, linear algebraic proof of the strong truncated Hamburger matrix moment problem in the nonsingular case. The results will be important in the study of the truncated univariate rational matrix moment problem.