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The inertia of Hermitian matrices with a prescribed $$2\times{}2$$ block decomposition. (English) Zbl 0756.15024
The main result of this paper gives a characterization of the inertia, $$\text{In}(H)=(\pi,\nu,*)$$, of a Hermitian matrix of the form $H=\left[{H_ 1\atop X^*} {X\atop H_ 2}\right]$ where $$r\leq\text{rank}(X)\leq R$$ and $$\text{In}(H_ i)=(\pi_ i,\nu_ i,*)$$, $$i=1,2$$, in terms of $$r$$, $$R$$, the dimensions of $$H_ i$$, $$i=1,2$$, and their inertia. As special cases of this result, a characterization of $$\text{In}(H)$$ is given when the restriction on $$\text{In}(H_ 2)$$ only, and on both $$\text{In}(H_ 1)$$ and $$\text{In}(H_ 2)$$ are relaxed. If no restrictions are imposed on $$\text{rank}(X)$$ and $$\text{In}(H_ i)$$, $$i=1,2$$, the result reduces to a previous one by the same authors.

##### MSC:
 15A42 Inequalities involving eigenvalues and eigenvectors 15A18 Eigenvalues, singular values, and eigenvectors 15A03 Vector spaces, linear dependence, rank, lineability
##### Keywords:
rank; block decomposition; inertia; Hermitian matrix
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##### References:
 [1] DOI: 10.1016/0024-3795(81)90175-0 · Zbl 0456.15011 · doi:10.1016/0024-3795(81)90175-0 [2] DOI: 10.1016/0024-3795(68)90009-8 · Zbl 0186.33704 · doi:10.1016/0024-3795(68)90009-8 [3] DOI: 10.1016/0024-3795(81)90174-9 · Zbl 0457.15012 · doi:10.1016/0024-3795(81)90174-9 [4] Perlis S., Theory of Matrices (1952) · Zbl 0046.24102
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