ON THE SCHUR COMPLEMENT

Suppose B is a nonsingular principal submatrix of an nxn matrix A. We define the Schur Complement of B in A, denoted by (A/B), as follows: Let A' be the matrix obtained from A by a simultaneous permutation of rows and columns which puts B into the upper left corner of A'. Then (A/B) = G -...

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Bibliographic Details
Main Author: Haynsworth,Emilie V
Format: Report
Language:English
Subjects:
DETERMINANTS(MATHEMATICS)
EIGENVECTORS
HERMITIAN MATRICES
INVARIANCE
MATRICES(MATHEMATICS)
PERMUTATIONS
SWITZERLAND
THEOREMS
Theoretical Mathematics
VECTOR SPACES
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Summary:Suppose B is a nonsingular principal submatrix of an nxn matrix A. We define the Schur Complement of B in A, denoted by (A/B), as follows: Let A' be the matrix obtained from A by a simultaneous permutation of rows and columns which puts B into the upper left corner of A'. Then (A/B) = G - D(B superscript -1)C. Schur proved that the determinant of A is the product of the determinants of any non-singular principal submatrix B with its Schur Complement. The inertia of an Hermitian matrix A is given by the ordered triplet, In A = (pi, nu, delta), where pi denotes the number of positive, nu the number of negative, and delta the number of zero roots of the Matrix A. In a previous paper it was shown that the inertia of an Hermitian matrix can be determined from that of any non-singular principal matrix together with that of its Schur complement. That is, if A is Hermitian and B is a non-singular principal submatrix of A, then In A = In B + In (A/B). This result is used to prove an extension of a theorem by Marcus. (Author)