k H and inner product H, which is assumed to be antilinear with respect to the second component. The generalized Schur complement s plays an important role in the representations for xd in many cases [9, 14, 15, 17]. Applied Mathematics and Computation 217 :18, 7531-7536. Identifying H and the space of continuous The generalized Schur complement in group inverses and in (k +1)-potent matrices 6 c) A− ∈ A{1,2}. ... 静 马, Triangular Schur Complement of Generalized Strictly Doubly Diagonally Dominant Matrices, Pure Mathematics, 10.12677/PM.2020.102016, 10, 02, (100-105), (2020). In addition, some spectral theory related to this complement is analyzed. Source Banach J. … (2011) A representation for the Drazin inverse of block matrices with a singular generalized Schur complement. (2011) ARE-type iterations for rational Riccati equations arising in stochastic control. In the beginning Schur complements were used in the theory of matrices. Let R ∈ L(X,H). M.G. Trapp [4] extended the notion of Schur complements of matrices to shorted operators in Hilbert space operators, and Trapp defined the generalized Schur complement by replacing the ordinary inverse with the generalized inverse. d) A #exists and A− = A . Article information. Krein [19] and W.N. If we assume that the generalized Schur complement s is invertible in Theorem 2.1 and Theorem 2.2, then we can prove the next result. Li [13] investigated a Corollary 2.4. In particular, it is proved that the Schur complement, if it exists, is an H‐matrix and the class to which the Schur complement belongs is studied. The following is the basic The other one is to develop a formula for any power of the generalized Schur complement of an idempotent partitioned matrix and then to characterize when this generalized Schur complement is a (k+1)-potent matrix. As is known, the Schur complements of diagonally dominant matrices are diagonally dominant; the same is true of doubly diagonally dominant matrices. [9] presented representations for the Drazin inverse of a 2£2 block matix under conditions which involve W = AAD+ADBCAD and that the generalized Schur complement is equal to 0. It is a well-known fact (see [5]) that S = D − CA−B is invariant under all choice of A− ∈ A{1} if and only if the above conditions a) and b) hold. The generalized Schur complement of A in M is defined to be M/A equals D minus CA** plus B, where A** plus is the Moore-Penrose inverse of A. Hartwig et al. Ander-son and G.E. The other one is to develop a formula for any power of the generalized Schur complement of an idempotent partitioned matrix and then to characterize when this generalized Schur complement … e) A− = A†. Extending the corresponding notion for matrices or bounded linear operators on a Hilbert space, we define a generalized Schur complement for a nonnegative linear operator mapping a linear space into its dual, and we derive some of its properties.