×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.