Generalized Inverse Formulas Using the Schur Complement

Block recursive computation of generalized inverses

“…Some of the equations (1), (2), (3) and (4) as well as the equations (1 k ) and (5) for square matrices are satisfied by the matrix X = A (α) , if A and input matrices in recursive calls of Algorithm 3.1 satisfy conditions imposed in [2,5,19,24]. However, this assumption is prohibitive.…”

“…However, this assumption is prohibitive. Only the equation (2) holds in general [5]. In this way, 4.…”

“…Similarly, Y. Wei [24] found the sufficient conditions for the Drazin inverse to be represented in the form (3.1). In the paper [9] these results are generalized to the weighted Moore-Penrose inverse and the weighted Drazin inverse of A. Burns et al [5] investigated {1}, {2}, {1, 3}, {1, 4} inverses as well as the Moore-Penrose inverse of A in the form (3.1). In [19] these results are extended to the set of polynomial matrices.…”

“…Subsequently, the Moore-Penrose inverse X = A † of A satisfies the set of the equations (1), (2), (3) and (4). The Drazin inverse X = A D of A satisfies the equations (1 k ), (2) and (5).…”

Generalized Schur complements of matrices and compound matrices

“…Introduction. Many important results for compound matrices and Schur complements of matrices have been obtained in [2,3,9,10]. Recently, Smith [11], Liu et al [5,6], Li et al [4] obtained some estimates of eigenvalues and singular values for Schur complements of matrices.…”