The -positive semidefinite matrices A such that are nonsingular

Abstract

This work focuses on the class of L -positive semidefinite matrices A such that A A ( L ) ( † ) − A ( L ) ( † ) A is nonsingular, where A ( L ) ( † ) is the generalized Bott-Duffin inverse of A with respect to a subspace L . We give some equivalent characterizations for the nonsingularity of A A ( L ) ( † ) − A ( L ) ( † ) A , A A ( L ) ( † ) + A ( L ) ( † ) A and I n − A ( A ( L ) ( † ) ) 2 A mainly by rank equalities and subspace operations, among which relations are pointed out. Meanwhile we show the inverses of their general versions in a decomposition form. In addition, the continuity of this class of matrices is discussed based on the continuity of the generalized Bott–Duffin inverse.