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A generalized Schur complement for nonnegative operators on linear spaces [PDF]
Banach Journal of Mathematical Analysis, 2018 Extending the corresponding notion for matrices or bounded linear operators on a Hilbert space we define a generalized Schur complement for a non-negative linear operator mapping a linear space into its dual and derive some of its properties.
L Klotz
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On the generalized Drazin inverse in Banach algebras in terms of the generalized Schur complement
Applied Mathematics and Computation, 2016zbMATH Open Web Interface contents unavailable due to conflicting licenses.
E Dopazo
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Block idempotent matrices and generalized Schur complement
Applied Mathematics and Computation, 2007Given a matrix partitioned into \(2\times 2\) blocks, where the left upper block is a nonsingular matrix denoted as \(A\). The paper deals with the idempotency of the generalized Schur complement with a generalized inverse \(A^{(2)}_{T,S}\), where \(A^{(2)}_{T,S}\) denotes the {2}-inverse of \(A\) with range \(T\) and null space \(S\).
Jinhua Zhou, Guorong Wang
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Some generalized inverses of partition matrix and quotient identity of generalized Schur complement
Applied Mathematics and Computation, 2008The authors initially extend the notion of the generalized Schur complement and study the expression of the Moore-Penrose inverse of \(2\times 2\) block matrices. Then, they propose expressions of the group inverse and the Drazin inverse for a block matrix under some conditions.
Xingping Sheng
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The Schur complement of a generalized M-matrix
Linear Algebra and Its Applications, 1972 exaly +3 more sources
General H-matrices and their Schur complements
Frontiers of Mathematics in China, 2014This paper presents an extensive fully theoretical study of properties of generalized \(H\)-matrices. It is known, that \(H\)-matrices can be divided into three disjoint sets -- the invertible class \(H^I\), the singular class \(H^S\) and the mixed class \(H^M\).
Zhang, Cheng-Yi +3 more
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A Schur complement approach to a general extrapolation algorithm
Linear Algebra and Its Applications, 2003 The authors use Schur complements and their properties to obtain various interpretations of the E-transformation. It is proved that ratios of determinants similar to those appearing in the E-transformation can be recursively computed by a triangular recursive scheme and that, reciprocally, quantities computed by such a scheme can be expressed as a ...
Claude Brezinski
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Generalized Inverse Formulas Using the Schur Complement
SIAM Journal on Applied Mathematics, 1974A formula for various generalized inverses of a partitioned complex matrix is established under certain general conditions. The use of this formula in obtaining the Moore–Penrose inverse of an arbitrary complex matrix is discussed.
Burns, Fennell +3 more
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A Generalization of the Schur Complement by Means of the Moore–Penrose Inverse
SIAM Journal on Applied Mathematics, 1974Suppose the complex matrix M is partitioned into a $2 \times 2$ array of blocks; let $M_{11} = A,M_{12} = B,M_{21} = C,M_{22} = D$. The generalized Schur complement of A in M is defined to be $M/A = D - CA^ + B$, where $A^ + $ is the Moore–Penrose inverse of A. The relationship of the ranks of M, A, and $M/A$ is determined.
Carlson, David +2 more
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Distributed Schur Complement Techniques for General Sparse Linear Systems
SIAM Journal on Scientific Computing, 1999Preconditioning techniques for solving general sparse linear systems on distributed memory environments are presented. Two of them are based on an approximate solution process for the global system exploiting approximate LU factorizations for diagonal blocks of the Schur complement, while another uses a sparse approximate-inverse technique to determine
Yousef Saad, Maria Sosonkina
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