An application of the separation theorem for hermitian matrices

Markham, T. L.

doi:10.1090/S0002-9939-1975-0364290-1

An application of the separation theorem for hermitian matrices
HTML articles powered by AMS MathViewer

by T. L. Markham PDF

Proc. Amer. Math. Soc. 47 (1975), 61-64 Request permission

Abstract:

Suppose $H$ is an $n \times n$ hermitian matrix over the complex field partitioned as $H = \left (\begin {smallmatrix}A&B\\B*&C\end {smallmatrix}\right )$, where $C$ is invertible. Using the separation theorem on eigenvalues of hermitian matrices, bounds are obtained for the eigenvalues of $(H/C) = A - B{C^{ - 1}}{B^ \ast }$ in terms of the eigenvalues of $H$ and $C$.

References

Douglas E. Crabtree and Emilie V. Haynsworth, An identity for the Schur complement of a matrix, Proc. Amer. Math. Soc. 22 (1969), 364–366. MR 255573, DOI 10.1090/S0002-9939-1969-0255573-1
Emilie V. Haynsworth, Determination of the inertia of a partitioned Hermitian matrix, Linear Algebra Appl. 1 (1968), no. 1, 73–81. MR 223392, DOI 10.1016/0024-3795(68)90050-5
Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1964. MR 0175290
Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Inc., Boston, Mass., 1964. MR 0162808
Luck J. Watford Jr., The Schur complement of a generalized $M$-matrix, Linear Algebra Appl. 5 (1972), 247–255. MR 309972, DOI 10.1016/0024-3795(72)90006-7