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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
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[1] DOI: 10.1016/0024-3795(81)90175-0 · Zbl 0456.15011 · doi:10.1016/0024-3795(81)90175-0
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