Matricial Gaussian quadrature rules: singular case

Let $L$ be a linear operator on univariate polynomials of bounded degree taking values in real symmetric matrices, whose moment matrix is positive semidefinite. Assume that $L$ admits a positive matrix-valued representing measure $\mu$. Any finitely atomic representing measure 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 that contains a prescribed atom with a prescribed rank of the corresponding mass, thereby generalizing our recent result, which addresses the same problem in the case where the moment matrix is positive definite. As a corollary, we obtain a constructive, linear-algebraic proof of the strong truncated Hamburger matrix moment problem.