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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

##### MSC:
 15B57 Hermitian, skew-Hermitian, and related matrices 15A18 Eigenvalues, singular values, and eigenvectors 15A63 Quadratic and bilinear forms, inner products
##### Keywords:
Hermitian matrices; inertia; Hermitian extension
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##### References:
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