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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 A E = 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. [ABSTRACT FROM AUTHOR]

The weighted spanning tree enumerator of a graph G with weighted edges is the sum of the products of edge weights over all the spanning trees in G. In the special case that all of the edge weights equal 1, the weighted spanning tree enumerator counts the number of spanning trees in G. The Weighted Matrix-Tree Theorem asserts that the weighted spanning tree enumerator can be calculated from the determinant of a reduced weighted Laplacian matrix of G. That determinant, however, is not always easy to compute. In this paper, we show how two well-known results from linear algebra, the Matrix Determinant Lemma and the method of Schur complements, can be used to elegantly compute the weighted spanning tree enumerator for several families of graphs. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let A i ( i = 1 , 2 , … , n ) be strict contractions on a Hilbert space H . The n × n operator matrix H n ( A 1 , A 2 , ⋯ , A n ) = ( ( I − A j ⁎ A i ) − 1 ) i , j = 1 n is called a Hua-type operator matrix. In this note, we mainly investigate some results which are related to the Hua-type operator matrix. We firstly give some equivalent conditions for the positivity of n × n operator matrices ( I − A j ⁎ A i ) i , j = 1 n . Then the equation min { ‖ H 2 ( A 1 , A 2 ) ‖ : ‖ A 1 ‖ < 1 , ‖ A 2 ‖ < 1 } = 2 is shown. In particular, some equivalent conditions for strict contractions A 1 and A 2 such that ‖ H 2 ( A 1 , A 2 ) ‖ = 2 are obtained. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: 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 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. [Copyright &y& Elsevier]

Abstract: 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 are sign regular of a certain order. In particular, some results for totally nonnegative and totally nonpositive matrices are provided as our corollaries. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: 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 eigenvalues with positive real part, and 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. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]