Generalized Schur Complement Research Articles

Some generalized inverses of partition matrix and quotient identity of generalized Schur complement

In this paper, we first define some generalized Schur complements, then we study the representation of M–P inverse M 1 † group inverse M 1 g and Drazin M 1 d for the partitioned matrix M 1 = A B C D . Based on this, we give three kind quotient identities of generalized Schur complement about the partitioned matrix K = A B E C D F G H L and develops the results in paper [M. Redivo-Zaglia, Pseudo-Schur complement and their applications, Appl. Num. Math. 50 (2004) 511–519; K. Jbilou, A. Messaoudi, K. Tabaâ, Some Schur complement identities and applications to matrix exatrapolation methods, Linear Algebra Appl. 392 (2004) 195–210]. In the end of the paper, we give the application of these generalized Schur complement in solution of linear equation, and give numerical example to show our results.

Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications

In this paper, we establish the formulas of the maximal and minimal ranks of the quaternion matrix expression C 4 - A 4 XB 4 where X is a variant quaternion matrix subject to quaternion matrix equations A 1 X = C 1 , XB 2 = C 2 , A 3 XB 3 = C 3 . As applications, we give a new necessary and sufficient condition for the existence of solutions to the system of matrix equations A 1 X = C 1 , XB 2 = C 2 , A 3 XB 3 = C 3 , A 4 XB 4 = C 4 , which was investigated by Wang [Q.W. Wang, A system of four matrix equations over von Neumann regular rings and its applications, Acta Math. Sin., 21(2) (2005) 323–334], by rank equalities. In addition, extremal ranks of the generalized Schur complement D - CA - B with respect to an inner inverse A − of A, which is a common solution to quaternion matrix equations A 1 X = C 1 , XB 2 = C 2 , are also considered. Some previous known results can be viewed as special cases of the results of this paper.

A note on the representations for the Drazin inverse of 2 × 2 block matrices

In 1979, Campbell and Meyer proposed the problem of finding a formula for the Drazin inverse of a 2 × 2 matrix M = A B C D in terms of its various blocks, where the blocks A and D are required to be square matrices. Special cases of the problems have been studied. In particular, a representation of the Drazin inverse of M, denoted by M D , has recently been obtained under the assumptions that C( I − AA D ) B = O and A( I − AA D ) B = O together with the condition that the generalized Schur complement D − CA D B be either nonsingular or zero. We derive an alternative representation for M D under the same assumptions, but with the condition on the Schur complement in the hypothesis replaced by the condition that R ( CAA D ) ⊂ N ( B ) ∩ N ( D ) , where R ( · ) and N ( · ) are the range and null space of a matrix.

The generalized Schur complement in group inverses and (k + 1)-potent matrices

In this article, two facts related to the generalized Schur complement are studied. The first one is to find necessary and sufficient conditions to characterize when the group inverse of a partitioned matrix can be expressed in the Schur form. 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. In addition, some spectral theory related to this complement is analyzed.

Block idempotent matrices and generalized Schur complement

In this paper, we study the idempotency of the generalized Schur complement on generalized inverse A T , S ( 2 ) , together with other familiar generalized inverses such as A M , N † , A †, A d and A g, defined from a partitioned idempotent matrix, which is a complementarity of [Jerzy K. Baksalary, Oskar Maria Baksalary, Tomasz Szulc, Properties of Schur complements in partitioned idempotent matrices, Lin. Alg. Appl. 379 (2004) 303–318; D.V. Ouellette, Schur complememnts and statistics, Lin. Alg. Appl. 36 (1981) 187–295].

Schur complements of matrices with acyclic bipartite graphs

Bipartite graphs are used todescribe the generalized Schur complements of real matrices having nosquare submatrix with twoor more nonzerodiagonals. For any matrix A with this property, including any nearly reducible matrix, the sign pattern of each generalized Schur complement is shown to be determined uniquely by the sign pattern of A. Moreover, if A has a normalized LU factorization A = LU, then the sign pattern of A is shown to determine uniquely the sign patterns of L and U, and (with the standard LU factorization) of L−1 and, if A is nonsingular, of U−1. However, if A is singular, then the sign pattern of the Moore-Penrose inverse U† may not be uniquely determined by the sign pattern of A. Analogous results are shown to hold for zero patterns.

Representations for the Drazin Inverse of a 2 x 2 Block Matrix

Two representations for the Drazin inverse of a $2\times2$ block matrix $M=[{A \atop C}\;{B \atop D}]$, where A and D are square matrices, in terms of the Drazin inverses of A and D have been recently developed under the assumptions that $C(I-AA^{D})=0$ and $(I-AA^{D})B=0$, and that the generalized Schur complement $D-CA^{D}B$ is either nonsingular or zero. These two representations of $M^{D}$ are extended to the case where $C(I-AA^{D})=0$ and $(I-AA^{D})B=0$ are substituted with $C(I-AA^{D})B=0$ and $A(I-AA^{D})B=0$. Moreover, upper bounds for the index of M are studied. Numerical examples are given to illustrate the new results.

Generalized Schur complements and oblique projections

Let S be a closed subspace of a Hilbert space H and A a bounded linear selfadjoint operator on H . In this note, we show that the existence of A-selfadjoint projections with range S is related to some properties of shorted operators, Schur complements (in Ando's generalization of the classical concept) and compressions.

The Moore–Penrose inverse of a partitioned nonnegative definite matrix

Consider an arbitrary symmetric nonnegative definite matrix A and its Moore–Penrose inverse A + , partitioned, respectively as A = E F F ′ H and A += G 1 G 2 G 2 ′ G 4 . Explicit expressions for G 1 , G 2 and G 4 in terms of E , F and H are given. Moreover, it is proved that the generalized Schur complement ( A +/ G 4)= G 1− G 2 G 4 + G 2 ′ is always below the Moore–Penrose inverse ( A / H ) + of the generalized Schur complement ( A / H )= E − FH + F ′ with respect to the Löwner partial ordering.

Problems and Techniques

This installment of Problems and Techniques features articles on extremal properties of the Schur complement of a matrix and the computation of superconvergent functionals from approximate solutions of partial differential equations. In the first article, Chi-Kwong Li and Roy Mathias describe extremal properties of the generalized Schur complement of a positive semidefinite matrix. The generalization refers to the use of a pseudoinverse when the matrix is not invertible. Using the extremal properties, the authors derive several operator and eigenvalue inequalities involving generalized Schur complements. While some of these results were known, the extremal principles simplify their analysis and also provide a technology for generating several new inequalities. The authors also provide an interesting historical perspective of the development and use of Schur complements. The use of the adjoint solutions has been suggested by several investigators as a means of constructing a posteriori error bounds for finite element solutions and their integral functionals. Niles Pierce and Michael Giles use this approach more generally, without reliance on the finite element method, to correct the leading order error term in integral functional estimates. This is a very powerful technique that can be used to construct superconvergent approximations of the integral functionals or to control an adaptive enrichment process. The authors indicate several problems where accuracy of an integral functional is of primary concern. In optimal design, for example, the adjoint solution is often needed as part of the optimization. The importance of this work was recognized when Niles Pierce received the Fox Prize in Numerical Analysis last June. I hope that you enjoy these articles as much as I did. Please join me in thanking our authors for their very interesting contributions.

Extremal Characterizations of the Schur Complement and Resulting Inequalities

Let \[ H=\pmatrix{H_{11}&H_{12}\cr H_{12}^*&H_{22}} \] be an $n\times n$ positive semidefinite matrix, where $H_{11}$ is $k\times k$ with $1 \le k < n$. The {\it generalized Schur complement} of $H_{11}$ in H is defined as $$ S(H) = H_{22} - H_{12}^*H_{11}^{\dag}H_{12}, $$ where $H_{11}^{\dag}$ is the Moore\,--Penrose generalized inverse of $H_{11}$. It has theextremal characterizations \[ S(H) = \max\left\{W: H - \pmatrix{0_k & 0 \cr 0 & W\cr} \ge 0, W \hbox{ is } (n-k) \times (n-k) \hbox{ Hermitian} \right\} \] and\rule[-4.75pt]{0pt}{0pt} $$ S(H) = \min \left\{[Z|I_{n-k}] H [Z|I_{n-k}]^*: Z \ \hbox{ is } \ (n-k) \times k \right\}. $$ \noindent These characterizations are used to deduce many old and new inequalities for Schur complements of positive semidefinite matrices. In many cases, stronger statements and shorter proofs can be obtained using the extremal characterizations.

Some inequalities on generalized Schur complements

This paper presents some inequalities on generalized Schur complements. Let A be an n×n (Hermitian) positive semidefinite matrix. Denote by A/α the generalized Schur complement of a principal submatrix indexed by a set α in A. Let A+ be the Moore–Penrose inverse of A and λ(A) be the eigenvalue vector of A. The main results of this paper are: 1.λ(A+(α′))⩾λ((A/α)+), where α′ is the complement of α in {1,2,…,n}.2.λ(Ar/α)⩽λr(A/α) for any real number r⩾1.3.(C*AC)/α⩽C*/αA(α′)C/α for any matrix C of certain properties on partitioning.

Some Löwner partial orders of Schur complements and Kronecker products of matrices

In this paper, we define generalized Schur complements of matrices. Further, combining Schur complements of matrices with Kronecker products of matrices, we study their Löwner partial order and obtain some important inequalities, that improve recent results.

Some inequalities for singular values and eigenvalues of generalized Schur complements of products of matrices

In this paper, using transformation of Schur complements of matrices and some estimates of eigenvalues of positive semidefinite Hermitian matrices, we obtain some inequalities of singular values for Schur complements of products of matrices.

The Extreme Points of Order Intervals of Positive Operators

Consider the order interval of operators [0, A] = { X|0 ≤ X ≤ A}. In finite dimensions (or if A is invertible) then the extreme points of [0, A] are the shorted operators (generalized Schur complements) of A. This is false in the general infinite dimensional case. We give an example arising from the discretization of the biharmonic equation. We also give necessary and sufficient conditions for an extreme point X 0 to be a short of A.

The Restricted Singular Value Decomposition: Properties and Applications

The restricted singular value decomposition (RSVD) is the factorization of a given matrix, relative to two other given matrices. It can be interpreted as the ordinary singular value decomposition with different inner products in row and column spaces. Its properties and structure are investigated in detail as well as its connection to generalized eigenvalue problems, canonical correlation analysis and other generalizations of the singular value decomposition. Applications that are discussed include the analysis of the extended shorted operator, unitarily invariant norm minimization with rank constraints, rank minimization in matrix balls, the analysis and solution of linear matrix equations, rank minimization of a partitioned matrix and the connection with generalized Schur complements, constrained linear and total linear least squares problems, with mixed exact and noisy data, including a generalized Gauss-Markov estimation scheme. Two constructive proofs of the RSVD in terms of other generalizations of the ordinary singular value decomposition are provided as well.

Positive solutions to [formula omitted

We study the positive (semidefinite) solutions to the matrix equation X = A− BX -1B ∗ under the assumption that A⩾0. We show that a positive solution exists if and only if a certain block tridiagonal operator is positive, in which case the solution is given by the shorted operator (or generalized Schur complement) of that operator.

Infinite networks and quadratic optimal control

A discrete-time quadratic optimal control problem is shown to be equivalent to an infinite cascade ofn-port electrical networks. Both problems can be expressed as a generalized Schur complement (or shorted operator) of a block tridiagonal operator. Using Maxwell's principle for resistive networks, a variational formulation of the cascade limit ofn-port networks is given. For one special case a formula for the solution of the quadratic control problem is given in terms of the geometric mean of operators.

On extending determinantal identities

A new principle for extending determinantal identities is established which generalizes Muir's classical law of extensible minors. The proof makes use of general elimination strategies and of generalized Schur complements. This principle allows either the list of columns or the list of rows extending the corresponding list in the given identity to depend on the latter. As applications of this technique Karlin's and a generalization of Sylvester's identity are derived.

An alternative approach to the parallel sum

The parallel sum of two positive operators on a Hilbert space H is defined by the formula: A: B = lim ε ↓ 0 a( A + B + ε I) −1 B. If the operator ( A + B) is invertible then this reduces to A: B = A( A + B) −1 B. The obvious generalization to non-invertible A + B is A(A + B) †B, where (A + B) † denotes the (possibly unbounded) Moore-Penrose pseudoinverse of ( A + B). This, however, does not work. In fact, from an abstract point of view, the operation ( A, B) ⇒ A: B is not purely algebraic. Nevertheless, we show that the parallel sum of A and B can be written as A:B¦ ker(A + B) ⊥ = A ~ (A + B) ~ −1 B ~¦ ker(A + B) ⊥ , where A ~ denotes an extension of A to the Hilbert space constructed by completing the inner product space H ker(A + B) with the inner product ( x, y) A + B = (( A + B) x, y). Both sides of the above equation vanish on ker( A + B). The above results depend on the infinite network interpretations of the shorted operator (or generalized Schur complement) due to C. A. Butler and the author.