ON THE SCHUR COMPLEMENT.
BASEL UNIV (SWITZERLAND) MATHEMATICS INST
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Suppose B is a nonsingular principal submatrix of an nxn matrix A. We define the Schur Complement of B in A, denoted by AB, 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 AB G - DB superscript -1C. 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 AB. This result is used to prove an extension of a theorem by Marcus. Author
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