The inertia of certain Hermitian block matrices
Hermitian matrices partitioned as \[ H=\left [\begin{matrix} H_1 & X_{12} & X_{13}\\ X^*_{12} & H_2 & 0\\ X^*_{13} & 0 & H_3 \end{matrix} \right] \] are considered. Sets of inertias of such matrices are characterized by a system of inequalities involving the orders of the blocks, the inertias of the blocks \(H_i\) and the ranks of the blocks \(X_{1j}\), \(j=2,3\). The main result generalizes a theorem of \textit{B. E. Cain} and \textit{E. Marques de Sá} [ibid. 160, 75-87 (1992; Zbl 0752.15002)] concerning partitioned matrices with null block \(H_3\). This main result is generalized to \(p\times p\) block decomposition of Hermitian matrices, \(p\geq 3\).
scientific article; zbMATH DE number 1154661