Some interlacing properties of the Schur complement of a Hermitian matrix. (English) Zbl 0765.15007

Smith, Ronald L.

Linear Algebra Appl. 177, 137-144 (1992).

The author presents the following result: — It is shown for a Hermitian matrix \(H\) with nonsingular principal submatrix \(A\), that the eigenvalues of the Moore-Penrose inverse of the Schur complement \((H/A)\) of \(A\) in \(H\) interlace the eigenvalues of the Moore-Penrose inverse of \(H\). Also, if \(H\) is semidefinite, then the eigenvalues of \((H/A)\) interlace the eigenvalues of \(H\).

Reviewer: S.Sridhar (Madras)

MSC:

15A42	Inequalities involving eigenvalues and eigenvectors
15B57	Hermitian, skew-Hermitian, and related matrices
15A09	Theory of matrix inversion and generalized inverses

Keywords:

Hermitian matrix; eigenvalues; Moore-Penrose inverse; Schur complement

References:

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