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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, $$p\geq 3$$.

##### MSC:
 15A45 Miscellaneous inequalities involving matrices 15A42 Inequalities involving eigenvalues and eigenvectors 15B57 Hermitian, skew-Hermitian, and related matrices
##### Keywords:
Hermitian block matrix; inertia; system of inequalities
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##### References:
 [1] Cain, B.E.; de Sá, E.Marques, The inertia of a Hermitian matrix having prescribed complementary principal submatrices, Linear algebra appl., 37, 161-171, (1981) · Zbl 0456.15011 [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 [3] Cain, B.E.; de Sá, E.Marques, The inertia of certain skew-triangular block matrices, Linear algebra appl., 160, 75-85, (1992) · Zbl 0752.15006 [4] Haynsworth, E.V.; Ostrowski, A.M., On the inertia of some classes of partitioned matrices, Linear algebra appl., 1, 299-316, (1968) · Zbl 0186.33704 [5] Perlis, S., Theory of matrices, (1952), Addison-Wesley Reading, Mass · Zbl 0046.24102 [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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