On the Perturbation of Schur Complements in Positive Semidefinite Matrices

This note gives perturbation bounds for the Schur complement of a positive definite matrix in a positive semidefinite matrix. This report is available by anonymous ftp from thales.cs.umd.edu in the directory pub/reports. y Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742. On the Perturbation of Schur Complements in Positive Semidefinite Matrices G. W. Stewart ABSTRACT This note gives perturbation bounds for the Schur complement of a positive definite matrix in a positive semidefinite matrix. Let A be a nonsingular matrix, and suppose A is embedded in a larger matrix P = ` A B 0 B C ' (here the prime does not denote a transpose). The Schur complement of A in P is the matrix S = C \Gamma B 0 A \Gamma1 B: Now suppose we replace P by ~ P = ` A+ E B 0 + F 0 B + F C +G ' where E, F , F 0 , and G are presumed small. Then up to first order terms, the Schur complement of A+ E in ~ P is ~ S ..