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Consider an operator A :H→K between Hilbert spaces and closed subspaces S ⊂ H and T ⊂ K. If there exist projections E on H and F on K such that R(E) =S, R(F) =T and AE=F∗A then A is called (S, T)-complementable. The origin of this notion comes from the idea of T. Ando of defining Schur complements in terms of operators. In this paper we present some characterizations of these triples (A, S, T) and applications to bilateral Schur complements and generalized Wiener-Hopf operators. Fil: Arias, Maria Laura. Universidad de Buenos Aires. Facultad de Ingeniería; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina Fil: Corach, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería; Argentina Fil: Maestripieri, Alejandra Laura. Universidad de Buenos Aires. Facultad de Ingeniería; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina

Based on the Schur complement, some upper bounds for the infinity norm of the inverse of generalized doubly strictly diagonally dominant matrices are obtained. In addition, it is shown that the new bound improves the previous bounds. Numerical examples are given to illustrate our results. By using the infinity norm bound, a lower bound for the smallest singular value is given.

On finite dimensional spaces, it is apparent that an operator is the product of two positive operators if and only if it is similar to a positive operator. Here, the class ${\mathcal L}^{+2}$ of bounded operators on separable infinite dimensional Hilbert spaces which can be written as the product of two bounded positive operators is studied. The structure is much richer, and connects (but is not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The spectral properties of operators in ${\mathcal L}^{+2}$ are developed, and membership in ${\mathcal L}^{+2}$ among special classes, including algebraic and compact operators, is examined., Comment: 33 pages. Dedicated to Henk de Snoo, on his 75th birthday. v3 corrects typos and includes some minor clarifications. To appear in Complex Analysis and Operator Theory

Three-dimensional finite-difference frequency-domain analyses of partially periodic photonic structures result in large-scale ill-conditioned linear systems. Due to the lack of efficient preconditioner and reordering scheme, existed general-purpose iterative and direct solvers are inadequate to solve these linear systems in time or memory. We propose an efficient direct solver to tackle this problem. By exploring the physical properties, the coefficient matrix structure, and hardware computing efficiency, we extend the concepts of grid geometry manipulation and multi-level Schur method to propose the Compressed Hierarchical Schur algorithm (CHiS). The proposed CHiS algorithm can use less memory and remove redundant computational workloads due to the homogeneity and periodicity of photonic structures. Moreover, CHiS relies on dense BLAS3 operations of sub-matrices that can be computed efficiently with strong scalability by the latest multicore processors or accelerators. The implementation and benchmarks of CHiS demonstrate promising memory usage, timing, and scalability results. The feasibility of future hardware acceleration for CHiS is also addressed using computational data. This high-performance analysis tool can improve the design and modeling capability for various photonic structures.

Given a bounded selfadjoint operator W on a Krein space H and a closed subspace S of H, the Schur complement of W to S is defined under the hypothesis of weak complementability. A variational characterization of the Schur complement is given and the set of selfadjoint operators W admitting a Schur complement with these variational properties is shown to coincide with the set of S-weakly complementable selfadjoint operators. Fil: Contino, Maximiliano. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería. Departamento de Matemáticas; Argentina Fil: Maestripieri, Alejandra Laura. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería. Departamento de Matemáticas; Argentina Fil: Marcantognini Palacios, Stefania Alma María. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina

In the first part of the paper, we deal with Euclidean Jordan algebraic generalizations of some results of Brualdi on inclusion regions for the eigenvalues of complex matrices using directed graphs. As a consequence, the theorems of Brauer–Ostrowski and Brauer on the location of eigenvalues are extended to the setting of Euclidean Jordan algebras. In the second part, motivated by the work of Li and Tsatsomeros on the class of doubly diagonally dominant matrices with complex entries and its subclasses, we present some inter-relations between the H -property, generalized strict diagonal dominance, invertibility, and strict double diagonal dominance in Euclidean Jordan algebras. In addition, we show that in a Euclidean Jordan algebra, the Schur complements of a strictly doubly diagonally dominant element inherit this property.

The external behavior of linear resistive circuits with terminals is characterized as a linear input-output map given by a weighted Laplacian matrix. Conditions are derived for shaping the external behavior of the circuit by interconnection with an additional resistive circuit. (C) 2010 Elsevier B.V. All rights reserved.

It is shown that the numerical rank of the off-diagonal blocks of certain Schur complements of matrices that arise from the finite-difference discretization of constant coefficient, elliptic PDEs in two spatial dimensions is bounded by a constant independent of the grid size. Moreover, in three-dimensional problems the Schur complements are shown to have off-diagonal blocks whose numerical rank is a slowly growing function.

Loo-Keng Hua showed some elegant matrix and determinant inequalities via a matrix identity and proved the positive semidefiniteness of a matrix involving the determinants of contractive matrices through group representation theory. His study was followed by M. Marcus, R. Bellman and T. Ando. The purpose of current paper is to revisit the Hua’s original work and the results of Marcus, Bellman and Ando with our comments, and to present analogs and extensions to their results.

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 ∈ SDn ∪ IDn 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 | J R + ( A ) | - | J R + α ( A ) | eigenvalues with positive real part and | J R - ( A ) | - | J R - α ( A ) | eigenvalues with negative real part.

New results on the patterns of linearly independent rows and columns among the blocks of a symplectic matrix are presented. These results are combined with the block structure of the inverse of a symplectic matrix, together with some properties of Schur complements, to give a new and elementary proof that the determinant of any symplectic matrix is +1. The new proof is valid for any field. Information on the zero patterns compatible with the symplectic structure is also presented.

It is well-known that the Schur complements of strictly diagonally dominant matrices are strictly diagonally dominant [D. Carlson, T. Markham, Schur complements of diagonally dominant matrices, Czech. Math. J. 29 (104) (1979) 246–251]; the same is true of generalized strictly diagonally dominant matrices [Jianzhou Liu, Yungqing Huang, Some properties on Schur complements of H-matrix and diagonally dominant matrices, Linear Algebra Appl. 389 (2004) 365–380]. In this paper, this result is extended to the block (strictly) diagonally dominant matrices and the generalized block (strictly) diagonally dominant matrices, that is, it is shown that the Schur complement of a block (strictly) diagonally dominant matrix is a block (strictly) diagonally dominant matrix and so is the Schur complement of a generalized block (strictly) diagonally dominant matrix.

application/pdf, By considering the Le Chatelier-Braun principle in thermodynamics for a system of linear equations, we derive a formula which describes how to calculate the inverse of principal submatrices of a real square matrix, based upon the inverse of the given matrix. This formula is related to the Schur complement, and is due to Duncan. Thus, the second term of Schur complement a la Duncan may be interpreted as the matrix of exact Le Chatelier-Braun effects. All proofs are simple and elementary.

application/pdf, In this research note, we extend the Duncan identity involving Schur complement to Banach spaces, and derive some propositions for the case where a given linear transformation is an M-operator. These results are natural generalizations of Le Chatelier-Samuelson principle discussed by Morishima, to Leontief models with an infinite number of commodities and processes.

We study the analytic properties of a matrix discrete system introduced in [7]. The singularity confinement for this system is shown to hold generically, i.e. in the whole space of parameters except possibly for algebraic subvarieties. This paves the way to a generalization of Painleve analysis to discrete matrix models., 15 pages. This second version is a more comprehensible version of our result stated in the first version

For three decades Schur complements have seen increasing applications in linear algebra, often as abstractions of Gaussian elimination. It is known that they obey certain nontrivial identities, such as Crabtree and Haynsworth's quotient property . We began this work asking if there were a theory for deciding their properties in general. Lambek's Categorial Grammar is a deductive system formalized in 1958 by Lambek as a mathematical foundation for a syntactic calculus of language. We show that Categorial Grammar gives a deductive system for deriving identities obeyed by LU-and UL-decompositions, Gaussian elimination, and Schur complements. At first impression this seems to be a strange result, connecting two unrelated topics. In retrospect, though, it is a consequence of the way both use quotients. It may have applications in developing grammatical formalisms and numerical algorithms.

In this work, we de ne the Dirichlet{to{Robin matrix associated with a Schr odinger type matrix on general networks, and we prove that it satis es the alternating property which is essential to characterize those matrices that can be the response matrices of a network. We end with some examples of the sign pattern behavior of the alternating paths.

This paper presents a simple unifying algorithm for solving systems of linear equations. Solving a system of linear equations will be interpreted as an interpolation problem. This new approach leads us to a general algorithm called the recursive interpolation algorithm ria , which includes the direct methods and some of the iterative methods. A version of the ria with pivoting strategy will be given. We will also show how to choose two free sets of parameters in the ria for recovering known direct methods. Other choices of these parameters yield some new methods.

Large-scale scientific applications and industrial simulations are nowadays fully integrated in many engineering areas. They involve the solution of large sparse linear systems. The use of large high performance computers is mandatory to solve these problems. The main topic of this research work was the study of a numerical technique that had attractive features for an efficient solution of large scale linear systems on large massively parallel platforms. The goal is to develop a high performance hybrid direct/iterative approach for solving large 3D problems. We focus specifically on the associated domain decomposition techniques for the parallel solution of large linear systems. We have investigated several algebraic preconditioning techniques, discussed their numerical be- haviours, their parallel implementations and scalabilities. We have compared their performances on a set of 3D grand challenge problems., La résolution de très grands systèmes linéaires creux est une composante de base algorithmique fondamentale dans de nombreuses applications scientifiques en calcul intensif. La résolution per- formante de ces systèmes passe par la conception, le développement et l'utilisation d'algorithmes parallèles performants. Dans nos travaux, nous nous intéressons au développement et l'évaluation d'une méthode hybride (directe/itérative) basée sur des techniques de décomposition de domaine sans recouvrement. La stratégie de développement est axée sur l'utilisation des machines mas- sivement parallèles à plusieurs milliers de processeurs. L'étude systématique de l'extensibilité et l'efficacité parallèle de différents préconditionneurs algébriques est réalisée aussi bien d'un point de vue informatique que numérique. Nous avons comparé leurs performances sur des systèmes de plusieurs millions ou dizaines de millions d'inconnues pour des problèmes réels 3D .

In this paper we study shorted operators relative to two different subspaces, for bounded operators on infinite dimensional Hilbert spaces. We define two notions of "complementability" in the sense of Ando for operators, and study the properties of the shorted operators when they can be defined. We use these facts in order to define and study the notions of parallel sum and subtraction, in this Hilbertian context., Facultad de Ciencias Exactas

ÖZET îstatistik'de bir matrisin rankı, determinantı, tersi ve genelleştirilmiş tersi önemli uygulamalara sahip olduğundan Schur komplementler matris teorisinde çok önemli bir kavramdır. Çalışmamızda Schur komplementlerde determinant, rank ve bir matrisin tersi ile ilgili bilinen sonuçları vererek bunları genelleştirilmiş inverslerde uygulamak için ne gibi uygulamalara sahip olmaları gerektiğini inceledik. Simetrik parçalanmış bir matrisin Inertia ' sının sol üst köşe veya sag alt köşedeki matrislerle nasıl ilişkili olduğunu, bölüm özelliğinin Schur komplementlerde nasıl ifade edildiğini vermeye çalıştık. îki pozitif semidefinite ? (pozitif definite) matrisin toplamlarının determinantının, sol üst köşelerindeki matrislerin determinantlarının toplamlarından büyük olduğunu Schur komplementleri kullanarak ispatladık. Schur komplementleri kullanarak optimal rank problemlerinin çözümlerinde gerekli özelliklerin neler olduğunu vermeye çalıştık. En son olarakda istatistik'de karşımıza çıkan parçalanmış matrislerin tersini, determinantını ve karekteristik köklerini Schur komplementleri kullanarak göstermeye çalıştık. VI SUMMARY In Statistics, since a rank of matrix, its inverse, determinant, generalized inverse have wide applications Schur complements in matrix theory is a very important concept. In this study, we give well known results related to a determinant, rank and inverse of a matrix in Schur complements and study on the properties which are necessary to apply these results mentioned above to generalized inverses. We study on how Inertia of a symmetric partitioned matrix is related to upper left corner or lower right corner matrices and try to give the quotient property in Schur complements. We proved that the determinant of addition of two positive semidefinite (or positive definite) matrices are greater than the addition of the determinants of submatrices in upper left corner matrix by using Schur complements. We study on the necessary properties for solutions of optimal rank problems by using Schur complements. Finally, we examined the inverse of a partitioned matrix and its determinants and its eigenvalues (characteristic roots) by using Schur complements. 127

The issue regarding Schur complements of sign regular matrices is rather subtle. It is known that the class of totally nonnegative matrices is not closed under arbitrary Schur complementation. In this paper, we demonstrate how Schur complements of sign regular matrices of order k are sign regular of a certain order. In particular, some results for totally nonnegative and totally nonpositive matrices are provided as our corollaries.

This paper presents an unified algorithm for solving systems of matrix and block linear equations. Solving a system of block linear equations will be interpreted as a matrix interpolation problem. This new approach gives us a general algorithm called the matrix recursive interpolation algorithm (MRIA), which includes block direct methods. The orthogonal version of the MRIA will be given and denoted by OMRIA. We will also show how to choose two free parameters in the MRIA for obtaining known and new block direct methods.

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.

We discuss a methodology to construct sparse approximations of Schur complements of two-by-two block matrices arising in Finite Element discretizations of partial differential equations. Earlier results from [2] are extended to more general symmetric positive definite matrices of two-by-two block form. The applicability of the method for general symmetric and nonsymmetric matrices is analysed. The paper demonstrates the applicability of the presented method providing extensive numerical experiments.

In the general linear model $\mathscr{E}(\mathbf{y}) = \mathbf{X\beta}$, the vector $\mathbf{A\beta}$ is estimable whenever there is a matrix $\mathbf{B}$ so that $\mathscr{E}(\mathbf{By}) = \mathbf{A\beta}$. Several characterizations of estimability are presented along with short easy proofs. The characterizations involve rank equalities, generalized inverses, Schur complements and partitioned matrices.

An inequality established by G. P. H. Styan (1973, Linear Algebra Appl.,6, 217–240) is on the Hadamard product and a correlation matrix. An inequality obtained by B.-Y. Wang and F. Zhang (1997, Linear and Multilinear Algebra, 43, 315–326) involves the Hadamard product and Schur complements. These two inequalities hold in the positive definite matrix case. Based on Albert's theorem, we present their extensions to cover the positive semidefinite matrix case. We also give relevant inequalities.