Schur complements and statistics

In this paper we discuss various properties of matrices of the type

$\text{S}\text{=}\text{H}\text{?}\text{GE}{}^{\mathrm{?1}}\text{F}$

, which we call the Schur complement of E in

$\text{A}\text{=}\text{E}\text{F}\text{G}\text{H}$

The matrix E is assumed to be nonsingular. When E is singular or rectangular we consider the generalized Schur complements $\text{S}\text{=}\text{H}\text{?}\text{GE}{}^{\mathrm{?}}\text{F}$, where E^? is a generalized inverse of E. A comprehensive account of results pertaining to the determinant, the rank, the inverse and generalized inverses of partitioned matrices, and the inertia of a matrix is given both for Schur complements and for generalized Schur complements. We survey the known results in a historical perspective and obtain several extensions. Numerous applications in numerical analysis and statistics are included. The paper ends with an exhaustive bibliography of books and articles related to Schur complements.     

Criteria and Schur complements of H-matrices

The purpose of this paper is twofold: We first present a sufficient condition for testing strictly generalized diagonally dominant matrices (i.e., H-matrices) and we claim that our criterion is superior to the existing ones. We then show that the proper subset of the H-matrices determined by the condition preserves the closure property under the Schur complement operation.     

A matrix inequality on Schur complements

We investigate a matrix inequality on Schur complements defined by {1}-generalized inverses, and obtain simultaneously a necessary and sufficient condition under which the inequality turns into an equality. This extends two existing matrix inequalities on Schur complements defined respectively by inverses and Moore-Penrose generalized inverses (see Wang et. al. [Lin. Alg. Appl., 302–303 (1999) 163–172] and Liu and Wang [Lin. Alg. Appl., 293(1999) 233–241]). Moreover, the non-uniqueness of {1}-generalized inverses yields the complicatedness of the extension.     

General H-matrices and their Schur complements

The definitions of θ-ray pattern proposed to establish some new results matrix and θ-ray matrix are firstly on nonsingularity/singularity and convergence of general H-matrices. Then some conditions on the matrix A ∈ C^n×n and nonempty α （n） = {1,2,... ,n} are proposed such that A is an invertible H-matrix if A（α） and A/α are both invertible H-matrices. Furthermore, the important results on Schur complement for general H-matrices are presented to give the different necessary and sufficient conditions for the matrix A E HM and the subset α C （n） such that the Schur complement matrix A/α∈ HI^n-｜α｜ or A/α ∈ Hn-｜α｜^M or A/α ∈ H^n-｜ α｜^S.     

Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices

In this paper, we prove that the diagonal-Schur complement of a strictly doubly diagonally dominant matrix is strictly doubly diagonally dominant matrix. The same holds for the diagonal-Schur complement of a strictly generalized doubly diagonally dominant matrix and a nonsingular H-matrix. We point out that under certain assumptions, the diagonal-Schur complement of a strictly doubly (doubly product) γ-diagonally dominant matrix is also strictly doubly (doubly product) γ-diagonally dominant. Further, we provide the distribution of the real parts of eigenvalues of a diagonal-Schur complement of H-matrix. We also show that the Schur complement of a γ-diagonally dominant matrix is not always γ-diagonally dominant by a numerical example, and then obtain a sufficient condition to ensure that the Schur complement of a γ-diagonally dominant matrix is γ-diagonally dominant.     

广义Schur补的广义逆

对四分块矩阵A=A(︿) A(︿,︿′)A(︿′,︿) A(︿′)来说 ,如果 A和 A(︿)都是非奇异的 ,则A- 1 (︿′) =(A/︿) - 1 ,这里 A/ ︿=A(︿′) -A(︿′,︿) A(︿) - 1 A(︿,︿′)是 A(︿)在 A中的 Schur补 .王伯英教授指出上述等式 ,对半正定的 Hermitian矩阵而言 ,一般也是不能推广到 Moore-Penrose逆上去的 .在某些限制条件下 ,我们证明了广义逆的主子矩阵与广义 Schur补的关系是密切的 ,它使经典结果成为特例     

Schur complements, Schur determinantal and Haynsworth inertia formulas in Euclidean Jordan algebras

In this article, we study the concept of Schur complement in the setting of Euclidean Jordan algebras and describe Schur determinantal and Haynsworth inertia formulas.     

Schur complements on Hilbert spaces and saddle point systems

For any continuous bilinear form defined on a pair of Hilbert spaces satisfying the compatibility Ladyshenskaya–Babušca–Brezzi condition, symmetric Schur complement operators can be defined on each of the two Hilbert spaces. In this paper, we find bounds for the spectrum of the Schur operators only in terms of the compatibility and continuity constants. In light of the new spectral results for the Schur complements, we review the classical Babušca–Brezzi theory, find sharp stability estimates, and improve a convergence result for the inexact Uzawa algorithm. We prove that for any symmetric saddle point problem, the inexact Uzawa algorithm converges, provided that the inexact process for inverting the residual at each step has the relative error smaller than 1/3. As a consequence, we provide a new type of algorithm for discretizing saddle point problems, which combines the inexact Uzawa iterations with standard a posteriori error analysis and does not require the discrete stability conditions.     

Schur complements and LDU decomposition in an additive category

This note establishes the LDU decomposition of a morphism, satisfying the usual sort of hypotheses, in an additive category The Schur complement, defined in terms of this decomposition satisfies a quotient formula. For an appropriate choice of category we get an application to matrices, namely that a set of commutators relative to the blocks of a matrix. A, satisfies the same relations relative to its LD. and U. The Schur complement is shown to describe a quotient or kernel action in the case of an m by n or n by m commutator of rank m linking two matrices, extending some spectral results of Haynsworth. Again a suitable category yields further results on the usual Schur complement. Some non-matrix (topological group) examples are mentioned.     

More results on Schur complements in Euclidean Jordan algebras

In a recent article Gowda and Sznajder (Linear Algebra Appl 432:1553–1559, 2010) studied the concept of Schur complement in Euclidean Jordan algebras and described Schur determinantal and Haynsworth inertia formulas. In this article, we establish some more results on the Schur complement. Specifically, we prove, in the setting of Euclidean Jordan algebras, an analogue of the Crabtree-Haynsworth quotient formula and show that any Schur complement of a strictly diagonally dominant element is strictly diagonally dominant. We also introduce the concept of Schur product of a real symmetric matrix and an element of a Euclidean Jordan algebra when its Peirce decomposition with respect to a Jordan frame is given. An Oppenheim type inequality is proved in this setting.     

The eigenvalue distribution on Schur complements of H-matrices

The paper studies the eigenvalue distribution of some special matrices. Tong in Theorem 1.2 of [Wen-ting Tong, On the distribution of eigenvalues of some matrices, Acta Math. Sinica (China), 20 (4) (1977) 273-275] gives conditions for an n × n matrix A ∈ SD_n ∪ ID_n to have |J_R+(A)| eigenvalues with positive real part, and |J_R-(A)| eigenvalues with negative real part. A counter-example is given in this paper to show that the conditions of the theorem are not true. A corrected condition is then proposed under which the conclusion of the theorem holds. Then the corrected condition is applied to establish some results about the eigenvalue distribution of the Schur complements of H-matrices with complex diagonal entries. Several conditions on the n × n matrix A and the subset α ⊆ N = {1, 2, … , n} are presented such that the Schur complement matrix A/α of the matrix A has eigenvalues with positive real part and eigenvalues with negative real part.     

On the infinitesimal limits of the Schur complements of tridiagonal matrices

In this paper we consider diagonally dominant tridiagonal matrices whose diagonals come from smooth functions. It is shown that the Schur complements or pivots that arise from Gaussian elimination of these matrices can be given point-wise limits on a grid as the matrices grow in size to infinity. Numerical results are presented to verify the theory.     

Algebraic multilevel preconditioning of finite element matrices using local Schur complements

We consider an algebraic multilevel preconditioning technique for SPD matrices arising from finite element discretization of elliptic PDEs. In particular, we address the case of non‐M matrices. The method is based on element agglomeration and assumes access to the individual element matrices. The left upper block of the considered multiplicative two‐level preconditioner is approximated using incomplete factorization techniques. The coarse‐grid element matrices are simply Schur complements computed from local neighbourhood matrices, i.e. small collections of element matrices. Assembling these local Schur complements results in a global Schur complement approximation that can be analysed by regarding (local) macro elements. These components, when combined in the framework of an algebraic multilevel iteration, yield a robust and efficient linear solver. The presented numerical experiments include also the Lamé differential equation for the displacements in the two‐dimensional plane‐stress elasticity problem. Copyright © 2005 John Wiley & Sons, Ltd.     

A scalable parallel factorization of finite element matrices with distributed Schur complements

We consider the parallel factorization of sparse finite element matrices on distributed memory machines. Our method is based on a nested dissection approach combined with a cyclic re‐distribution of the interface Schur complements. We present a detailed definition of the parallel method, and the well‐posedness and the complexity of the algorithm are analyzed. A lean and transparent functional interface to existing finite element software is defined, and the performance is demonstrated for several representative examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Diagonal dominance, Schur complements and some classes of H-matrices and P-matrices

We analyze a class of matrices generalizing strictly diagonally dominant matrices and included in the important class of H-matrices. Adequate pivoting strategies and the corresponding Schur complements are studied. A new class of matrices with all their principal minors positive is presented.     

Multigrid transfers for nonsymmetric systems based on Schur complements and Galerkin projections

A framework is proposed for constructing algebraic multigrid transfer operators suitable for nonsymmetric positive definite linear systems. This framework follows a Schur complement perspective as this is suitable for both symmetric and nonsymmetric systems. In particular, a connection between algebraic multigrid and approximate block factorizations is explored. This connection demonstrates that the convergence rate of a two‐level model multigrid iteration is completely governed by how well the coarse discretization approximates a Schur complement operator. The new grid transfer algorithm is then based on computing a Schur complement but restricting the solution space of the corresponding grid transfers in a Galerkin‐style so that a far less expensive approximation is obtained. The final algorithm corresponds to a Richardson‐type iteration that is used to improve a simple initial prolongator or a simple initial restrictor. Numerical results are presented illustrating the performance of the resulting algebraic multigrid method on highly nonsymmetric systems. Copyright © 2013 John Wiley & Sons, Ltd.     

关于广义Muirhead平均的Schur幂凸性

对广义Muirhead平均的Schur-幂凸性进行了讨论,给出了判定Muirhead平均的Schur-幂凸性的充要条件.结果改进了Chu和Xia在相关文献中的主要结果,Chu和Xia的结果是结果的特例.