×

zbMATH — the first resource for mathematics

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

MSC:
15B57 Hermitian, skew-Hermitian, and related matrices
15A42 Inequalities involving eigenvalues and eigenvectors
15A18 Eigenvalues, singular values, and eigenvectors
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cottle, Richard W.; Ferland, Jacques A., Matrix-theoretic criteria for the quasi-convexity and pseudo-convexity of quadratic functions, Linear algebra appl., 5, 123-136, (1972) · Zbl 0241.90045
[2] Dancis, Jerome, On the inertias of symmetric matrices and bounded self-adjoint operators, Linear algebra appl., 105, 67-75, (1988) · Zbl 0644.15009
[3] Dancis, Jerome, The possible inertias for a Hermitian matrix and its principal submatrices, Linear algebra appl., 85, 121-151, (1987) · Zbl 0614.15011
[4] Dancis, Jerome; Davis, Chandler, An interlacing theorem for eigenvalues of self-adjoint operators, Linear algebra appl., 88/89, 117-122, (1987) · Zbl 0621.47017
[5] Dyn, Nira; Ferguson, Warren E., The numerical solution of equality-constrained quadratic programming problems, Math. comp., 41, 165-170, (1983) · Zbl 0527.49030
[6] Gohberg, Israel; Goldberg, Seymour, Basic operator theory, (1981)
[7] Hadley, G., Nonlinear and dynamic programming, (1964), Addison-Wesley · Zbl 0179.24601
[8] Haynsworth, E.V., Determination of the inertia of a partitioned Hermitian matrix, Linear algebra appl., 1, 73-81, (1968) · Zbl 0155.06304
[9] Haynsworth, E.V.; Ostrowski, A.M., On the inertia of some classes of partitioned matrices, Linear algebra appl., 1, 299-316, (1968) · Zbl 0186.33704
[10] Martos, B., Subdefinite matrices and quadratic forms, SIAM J. appl. math., 17, 1215-1223, (1969) · Zbl 0186.34201
[11] Martos, B., Oper. res., 19, 87-97, (1971)
[12] Molinari, B.P., The stabilizing solution of the algebraic Riccati equation, SIAM J. control, 11, 267-271, (1973) · Zbl 0254.34006
[13] George Weiss, private communication, 1984
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.