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Necessary and sufficient conditions for the Stieltjes moment problem to have a unique solution and for the Hamburger moment problem with the same moments to have infinitely many solutions were obtained in Hamburger's papers on the classical moment problem. In this paper, we obtain an analog of Hamburger's criterion for the matrix moment problem. [ABSTRACT FROM AUTHOR]

We introduce a simultaneous decomposition for a matrix triplet ( A, B, C), where A=± A and (⋅) denotes the conjugate transpose of a matrix, and use the simultaneous decomposition to solve some conjectures on the maximal and minimal values of the ranks of the matrix expressions A− BXC±( BXC) with respect to a variable matrix X. In addition, we give some explicit formulas for the maximal and minimal values of the inertia of the matrix expression A− BXC−( BXC) with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression D− CXC subject to Hermitian solutions of a consistent matrix equation AXA= B, as well as the extremal ranks and inertias of the Hermitian Schur complement D− B A B with respect to a Hermitian generalized inverse A of A. Various consequences of these extremal ranks and inertias are also presented in the paper. [ABSTRACT FROM AUTHOR]