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The inertia of certain Hermitian block matrices. (English) Zbl 0929.15019
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 B. E. Cain and 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, \(p\geq 3\).

15A45 Miscellaneous inequalities involving matrices
15A42 Inequalities involving eigenvalues and eigenvectors
15B57 Hermitian, skew-Hermitian, and related matrices
Full Text: DOI
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[2] Cain, B.E.; de Sá, E.Marques, The inertia of Hermitian matrices with a prescribed 2 × 2 block decomposition, Linear and multilinear algebra, 31, 119-130, (1992) · Zbl 0756.15024
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[6] de Sá, E.Marques, On the inertia of sums of Hermitian matrices, Linear algebra appl., 37, 143-159, (1981) · Zbl 0457.15012
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