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The possible inertias for a Hermitian matrix and its principal submatrices. (English) Zbl 0614.15011
Let \(H_{ii}\) be \(n_ i\times n_ i\) Hermitian matrices, \(i=1,2,...,m\), \((\pi_ i,\nu_ i,\delta_ i)=In H_{ii}\), the inertia of \(H_{ii}\) where \(\pi_ i\), \(\nu_ i\), \(\delta_ i\) are the numbers of positive, negative and zero eigenvalues of \(H_{ii}\) correspondingly, \(n=\sum^{m}_{i=1}n_ i\). The author calls an \(n\times n\)-Hermitian matrix H a Hermitian extension of the \(H_{ii}'s\) if the \(H_{ii}'s\) are the block diagonals of H. The objective of the paper is the study of relations between \((\pi_ i,\nu_ i,\delta_ i)\) and In H, the inertia of H. Examples of the results proved in the paper: 1) Given nonnegative integers \(\pi\), \(\nu\), \(\delta\) such that \(\pi +\nu +\delta =\sum (\pi_ i+\nu_ i+\delta_ i)\); a Hermitian extension H of \(\{H_{ii}\}\) such that Ker \(H\supset \oplus Ker H_{ii}\) and In H\(=(\pi,\nu,\delta)\) exists iff \(\delta\geq \sum \delta_ i\), \(\pi\geq \max \pi_ i\), \(\nu\geq \max \nu_ i\) (corollary 1.1); 2) A Hermitian extension H of \(\{H_{ii}\}\) such that Ker H/C\({}^{n_ i}=\{\bar O\}\) where \(C^{n_ i}\) is the domain of \(H_{ii}\), \(i=1,2,...,m\) exists iff \(\pi \geq \max (\pi_ i+\delta_ i)\), \(\nu \geq \max (\nu_ i+\delta_ i)\) (corollary 1.2).
Reviewer: B.Reichstein

15B57 Hermitian, skew-Hermitian, and related matrices
15A18 Eigenvalues, singular values, and eigenvectors
15A63 Quadratic and bilinear forms, inner products
Full Text: DOI
[1] de Sá, E.Marques, On the inertia of sums of Hermitian matrices, Linear algebra appl., 37, 143-159, (1981) · Zbl 0457.15012
[2] 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
[3] Cain, B.E., The inertia of a Hermitian matrix having prescribed diagonal blocks, Linear algebra appl., 37, 173-180, (1981) · Zbl 0462.15009
[4] J. Dancis, On the inertias of symmetric matrices and bounded self-adjoint operators, to appear. · Zbl 0644.15009
[5] Johnson, C.R.; Rodman, L., Inertia possibilities for completions of partial Hermitian matrices, Linear and multilinear algebra, 16, 179-195, (1984) · Zbl 0548.15020
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