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Several consequences of an inertia theorem. (English) Zbl 0712.15021
The main result of the paper is to show that if \[ H=\left[ \begin{matrix} H_ 1\\ A^*\end{matrix} \quad \begin{matrix} A\\ H_ 2\end{matrix} \right], \] where \(H_ 1\) and \(H_ 2\) are \(m\times m\) and \(r\times r\) positive semidefinite Hermitian matrices, then \(\pi =m-\dim (Ker H_ 1\cap Ker A)\), \(\nu =r-\dim (Ker H_ 2\cap Ker A^*)\), and Ker H\(=(Ker H_ 1\cap Ker A)\oplus (Ker H_ 2\cap Ker A^*)\), where \(\pi\) and \(\nu\) denote the number of positive and negative eigenvalues of H.
The result above is used to show that
a) a specific type of matrix, which arises in quadratic programming, is invertible,
b) certain real Hamiltonian matrices, associated with the algebraic Riccati equation, have no pure imaginary eigenvalues.
The main result is also extended to the case when H is a self-adjoint operator on an infinite dimensional space.
Reviewer: M.E.Sezer

15B57 Hermitian, skew-Hermitian, and related matrices
15A42 Inequalities involving eigenvalues and eigenvectors
15A18 Eigenvalues, singular values, and eigenvectors
Full Text: DOI
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