
Accession Number : AD0673276
Title : ON THE SCHUR COMPLEMENT.
Descriptive Note : Mathematical notes,
Corporate Author : BASEL UNIV (SWITZERLAND) MATHEMATICS INST
Personal Author(s) : Haynsworth,Emilie V.
Report Date : JUN 1968
Pagination or Media Count : 18
Abstract : 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 nonsingular 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 nonsingular principal matrix together with that of its Schur complement. That is, if A is Hermitian and B is a nonsingular 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)
Descriptors : (*MATRICES(MATHEMATICS), PERMUTATIONS), DETERMINANTS(MATHEMATICS), VECTOR SPACES, INVARIANCE, THEOREMS, SWITZERLAND
Subject Categories : Theoretical Mathematics
Distribution Statement : APPROVED FOR PUBLIC RELEASE