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In this paper, we obtain a new estimate for the (product) γ -diagonally dominant degree of the Schur complement of matrices. As applications we discuss the localization of eigenvalues of the Schur complement and present several upper and lower bounds for the determinant of strictly γ -diagonally dominant matrices, which generalizes the corresponding results of Liu and Zhang (SIAM J. Matrix Anal. Appl. 27 (2005) 665-674). [ABSTRACT FROM AUTHOR]

The present study proves that TT is an idempotent operator if and only if R(I−T∗)⊕R(T)=X{\mathcal{ {\mathcal R} }}\left(I-{T}^{\ast })\oplus {\mathcal{ {\mathcal R} }}\left(T)={\mathcal{X}} and (T∗T)†=(T†)2T{\left({T}^{\ast }T)}^{\dagger }={\left({T}^{\dagger })}^{2}T. Based on the equivalent conditions of an idempotent operator and related results, it is possible to obtain an explicit formula for the Moore-Penrose inverse of 2-by-2 block idempotent operator matrix. For the 2-by-2 block operator matrix, Schur complements and generalized Schur complement are well known and studied. The range inclusions of operators and idempotency of operators are used to obtain new conditions under which we can compute the Moore-Penrose inverse of Schur complements and generalized Schur complements of operators.

The theory of Schur complement is an important research area with numerous applications in scientific computing and statistics. This paper proves that the Schur complements and the diagonal-Schur complements of CKV-type matrices are CKV-type matrices under certain conditions. [ABSTRACT FROM AUTHOR]

In this paper, we present a constructive proof of Popescu’s Fejér-Riesz theorem for non-commuting polynomials representing nonnegative “multi-Toeplitz” operators. [ABSTRACT FROM AUTHOR]

Let A and B be two M-matrices, A - 1 be the inverse of A, and τ (B ∘ A - 1) be the minimum eigenvalue of the Hadamard product of B and A - 1 . Firstly, by using the theories of Schur complements, a lower bound of the main diagonal entries of A - 1 is derived and used to present two types of lower bounds of τ (B ∘ A - 1) . Secondly, in order to obtain bigger lower bounds of τ (B ∘ A - 1) , two types of lower bounds of τ (B ∘ A - 1) with non-negative parameters are constructed. Thirdly, by finding the optimal values of parameters, two preferable lower bounds of τ (B ∘ A - 1) are yielded. Finally, numerical examples show the effectiveness of the new methods. [ABSTRACT FROM AUTHOR]

It is shown in this paper that the Schur complements and the diagonal-Schur complements of DZ-type matrices are DZ-type matrices under some conditions. A numerical example for solving the linear equations with the coefficient matrix being a DZ-type matrix is given to show that the Schur-based Gauss–Seidel iteration method and the Schur-based conjugate gradient method can compute out the solution faster. [ABSTRACT FROM AUTHOR]

The four families of matrix orthogonal polynomials are considered arising in the truncated Hausdorff matrix moment (THMM) problem. Two of those families are associated with an odd number of moments and the other two with an even number of moments. The three-term recurrence relations associated with these four families are investigated. Certain explicit formulas are presented relating the three-term recurrence relation coefficients to the Dyukarev-Stieltjes parameters, the Schur complements and the orthogonal matrix polynomials associated with the THMM problem. The matrix version of the J-fraction is presented for the corresponding four extremal solutions of the THMM problem. [ABSTRACT FROM AUTHOR]

Based on the Schur complements, two upper bounds for the infinity norm of the inverse of doubly strictly diagonally dominant (DSDD) matrices are presented. As applications, an error bound for linear complementarity problems of DB-matrices and a lower bound for the smallest singular value of matrices are given. [ABSTRACT FROM AUTHOR]

We introduce a new and generic approximation to Schur complements arising from inf-sup stable mixed finite element discretizations of self-adjoint multiphysics problems. The approximation exploits the discretization mesh by forming local, or element, Schur complements of an appropriate system and projecting them back to the global degrees of freedom. The resulting Schur complement approximation is sparse, has low construction cost (with the same order of operations as assembling a general finite element matrix), and can be solved using off-the-shelf techniques, such as multigrid. Using results from saddle point theory, we give conditions such that this approximation is spectrally equivalent to the global Schur complement. We present several numerical results to demonstrate the viability of this approach on a range of applications. Interestingly, numerical results show that the method gives an effective approximation to the nonsymmetric Schur complement from the steady state Navier--Stokes equations. [ABSTRACT FROM AUTHOR]

New bounds for the doubly diagonally dominant degree of the Schur complement of strictly doubly diagonally dominant (SDD) matrices are derived and proved to be better than those in Liu et al. (Linear Algebra Appl 437:168–183, 2012). As applications, a new distribution of the eigenvalues and two new infinity norm bounds for the Schur complements of SDD matrices are obtained. Finally, numerical examples are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

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 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 L + 2 are developed, and membership in L + 2 among special classes, including algebraic and compact operators, is examined. [ABSTRACT FROM AUTHOR]

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]

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.

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]

The importance of Schur-complement-based preconditioners is well established for classical saddle point problems in |$\mathbb{R}^N \times \mathbb{R}^M$|⁠. In this paper we extend these results to multiple saddle point problems in Hilbert spaces |$X_1\times X_2 \times \cdots \times X_n$|⁠. For such problems with a block tridiagonal Hessian and a well-defined sequence of associated Schur complements, sharp bounds for the condition number of the problem are derived, which do not depend on the involved operators. These bounds can be expressed in terms of the roots of the difference of two Chebyshev polynomials of the second kind. If applied to specific classes of optimal control problems the abstract analysis leads to new existence results as well as to the construction of efficient preconditioners for the associated discretized optimality systems. [ABSTRACT FROM AUTHOR]

Constructive algorithms, not even requiring 2x2 eigencalculations, are provided for finding the entries of the positive definite factor in the polar decomposition of matrices in many groups. These groups include the indefinite orthogonal groups of signature (1, n -- 1) nd (n -- 1, 1) and fifteen groups preserving certain bilinear forms in dimension four. The Lorentz group belongs to both classes. Some of these algorithms extend to the indefinite orthogonal groups of arbitrary signature with nominal additional work. These procedures are then used to find quaternionic representations for the four dimensional groups mentioned above, analogous to the representation of the special orthogonal group via a pair of unit quaternions. A key ingredient is a characterization of positive definite matrices in these groups. Two algorithms are proposed for the Lorentz group. The former also works for the group whose signature is (n -- 1, 1) and (1, n -- 1). The second enables (and is aided by) the inversion of the double covering of the Lorentz group by SL(2, C). A key observation is that the inversion of the covering map, when the target is a positive definite matrix, can be achieved essentially by inspection as we demonstrate. For the group whose signature matrix is I2,2 a completion procedure based on the aforementioned characterization of positivity leads to yet another algorithm for the computation of the polar decomposition. For the other four dimensional groups, explicit isomorphisms provided by quaternion algebra lead to methods for the polar decomposition. As byproducts we give a simple proof of the fact that positive definite matrices in each of these groups belong to the connected component of the identity, find an explicit expression for their logarithm, and provide a characterization of the symmetric matrices in the connected component of the identity of two of these groups in terms of their preimages in the corresponding covering group. [ABSTRACT FROM AUTHOR]

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. [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]

In this paper, two accelerated divide-and-conquer (ADC) algorithms are proposed for the symmetric tridiagonal eigenvalue problem, which cost O(N2r) flops in the worst case, where N is the dimension of the matrix and r is a modest number depending on the distribution of eigenvalues. Both of these algorithms use hierarchically semiseparable (HSS) matrices to approximate some intermediate eigenvector matrices, which are Cauchy-like matrices and are off-diagonally low-rank. The difference of these two versions lies in using different HSS construction algorithms, one (denoted by ADC1) uses a structured low-rank approximation method and the other (ADC2) uses a randomized HSS construction algorithm. For the ADC2 algorithm, a method is proposed to estimate the off-diagonal rank. Numerous experiments have been carried out to show their stability and efficiency. These algorithms are implemented in parallel in a shared memory environment, and some parallel implementation details are included. Comparing the ADCs with highly optimized multithreaded libraries such as Intel MKL, we find that ADCs could be more than six times faster for some large matrices with few deflations. [ABSTRACT FROM AUTHOR]

This paper describes a multilevel preconditioning technique for solving sparse symmetric linear systems of equations. τhis "Multilevel Schur Low-Rank" (MSLR) preconditioner first builds a tree structure τ based on a hierarchical decomposition of the matrix and then computes an approximate inverse of the original matrix level by level. Unlike classical direct solvers, the construction of the MSLR preconditioner follows a top-down traversal of τ and exploits a low-rank property that is satisfied by the difference between the inverses of the local Schur complements and specific blocks of the original matrix. A few steps of the generalized Lanczos tridiagonalization procedure are applied to capture most of this difference. Numerical results are reported to illustrate the efficiency and robustness of the MSLR preconditioner with both two- and three-dimensional discretized PDE problems and with publicly available test problems. [ABSTRACT FROM AUTHOR]

As is well known, the diagonal-Schur complements of strictly diagonally dominant matrices are strictly diagonally dominant. In this paper, we verify the block diagonal-Schur complements of I-block strictly doubly diagonally dominant matrices are I-block strictly doubly diagonally dominant matrices, the same is true for II-block strictly doubly diagonally dominant matrices. The theoretical analysis is supported by numerical experiments. [ABSTRACT FROM AUTHOR]

The Schur complement structure with respect toprincipal submatrices, in hollow, symmetric nonnegative matrices is investigated, with an emphasis on such matrices that have only two nonpositive eigenvalues. It is shown that a wide family of such Schur complements simply follows a unique and surprising structure that can be fully described in a graph theoretical language, and is predictable from the entries. For larger numbers of nonpositive eigenvalues, conjectures regarding connections to polyhedra are also presented, and proved in a special case. Relations to copositive matrices and Morishima matrices are described as well. [ABSTRACT FROM AUTHOR]

When the low-cost inertial measurement units are used in actual vehicle localization, the measurements suffer from large noise density and unstable bias. Therefore, the integral gyroscope error becomes larger, which results in visual-inertial odometry system degradation or even failure. To this end, this paper proposes a robust visual-inertial odometry method based on a double-stage Kalman filter for a low-cost visual-inertial system, which consists of two Kalman filters. The first filter is a complementary Kalman filter, which uses the accelerometer to correct the gyroscope bias, and then an accurate initial pose estimation is calculated. The second filter is a multi-state observation-constrained Kalman filter, in which the re-projection error of features is calculated based on the multi-state observation constraint strategy to update the system states. Additionally, a Schur complement model is used for the sliding window to marginalize the oldest camera pose of the system states, avoiding the loss of associated information between images and improving the accuracy of the camera pose. Finally, the EuRoC dataset and a homemade low-cost visual-inertial hardware system are used to evaluate the performance of the proposed algorithm. The results show that the accuracy of the low-cost gyroscope bias estimation will decrease when the visual observation is inaccurate in the classic VIO, the proposed algorithm corrects the gyroscope bias with accelerometer measurements, which significantly improves the accuracy and robustness of the vehicle pose estimation when low-cost inertial measurement units are used. [ABSTRACT FROM AUTHOR]

In this work, we define the Dirichlet-to-Robin matrix associated with a Schrödinger type matrix on general networks, and we prove that it satisfies 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. [ABSTRACT FROM AUTHOR]

In this paper, aiming at solving the bidiagonal SVD problem, a classical divide-andconquer (DC) algorithm is modified, which needs to compute the SVD of broken arrow matrices by solving secular equations. The main cost of DC lies in the updating of singular vectors, which involves two matrix-matrix multiplications. We find that the singular vector matrices of a broken arrow matrix are Cauchy-like matrices and have an off-diagonal low-rank property, so they can be approximated efficiently by hierarchically semiseparable (HSS) matrices. Hereby, by using the HSS techniques, the complexity of computing singular vectors can be reduced significantly. An accelerated DC algorithm is proposed, denoted by ADC. Furthermore, we use a structured low-rank approximation method to construct these HSS approximations. Numerous experiments show ADC is both fast and numerically stable. When dealing with large matrices with few deflations, ADC can be 3x faster than DC in the optimized LAPACK libraries such as Intel MKL without any degradation in accuracy. These techniques can be used to similarly solve the symmetric tridiagonal eigenvalue problem. [ABSTRACT FROM AUTHOR]

Abstract: We show pivoting strategies such that the radii of the Geršgorin circles of the Schur complements through Gaussian elimination with these pivoting strategies reduce their length. The results will be illustrated with classes of matrices important in many applications. [Copyright &y& Elsevier]

The first time I met George Styan was in July 2004 in Lisbon when he was on his way to the 11th ILAS Conference in Coimbra. But George had already been in Portugal before and I learned how much he was fond of Conventual, a very fine and nice old style restaurant in Lisbon. Then I also learned that George really is an appreciator of good food and a very well-educated wine drinker. With this detail in common it was really easy to become a good friend with George. Since then we met a number of times, the most significant of which was at the time of the 17th IWMS held in Tomar, Portugal, in 2008. Before this event, during a short stay of George and Evelyn in Lisbon, we had the opportunity to go to some nice spots like Sintra and to hang around a few nice places near Lisbon and even to attend a Leonard Cohen concert, together with some friends. Actually, even more than good food and a good wine, and more than a good mathematical challenge, George enjoys the company of his family and his friends. We may even say that more than Mathematics, it is his family and his friends that play and have always played a central role in his life. Everybody knows well how much he cares about Evelyn, the great woman behind the great man, and also everybody knows the looks in George's face when he meets the ones he cares about. Inevitably, besides addressing some of George's honors and also his scientific work and his interest in mathematics related stamps, it is based on a number of pictures, either taken by the author or by other friends and a couple of them even taken by George himself, that this little contribution to the celebration of George Styan's 75th birthday will be indeed more a celebration of the way George enjoys and nurtures the company of the ones he loves. [ABSTRACT FROM AUTHOR]

In a recent article Gowda and Sznajder (Linear Algebra Appl 432:1553-1559, ) 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. [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]

It is shown that the numerical rank of the off-diagonal blocks of certain Schur complements of matrices that arise from the finitedifference 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. [ABSTRACT FROM AUTHOR]

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: 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. [Copyright &y& Elsevier]

Canonical factorization of a rational matrix function on the unit circle is described explicitly in terms of a stabilizing solution of a discrete algebraic Riccati equation using a special state space representation of the symbol. The corresponding Riccati difference equation is also discussed. [ABSTRACT FROM AUTHOR]

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]

The aim of this work is to generalize the notions of Schur complements and shorted operators to Krein spaces. Given a (bounded) J-selfadjoint operator A (with the unique factorization property) acting on a Krein space $${\mathcal{H}}$$ and a suitable closed subspace $${\mathcal{S}}$$ of $${\mathcal{H}}$$ , the Schur complement $$A_{/[\mathcal{S}]}$$ of A to $${\mathcal{S}}$$ is defined. The basic properties of $$A_{/[\mathcal{S}]}$$ are developed and different characterizations are given, most of them resembling those of the shorted of (bounded) positive operators on a Hilbert space. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new method for defining preconditioners for the iterative solution of a system of linear equations. By generalizing the class of modified preconditioners (e.g. MILU), the interface probing, and the class of preconditioners related to the Frobenius norm minimization (e.g. FSAI, SPAI) we develop a toolbox for computing preconditioners that are improved relative to a given small probing subspace. Furthermore, by this MSPAI (modified SPAI) probing approach we can improve any given preconditioner with respect to this probing subspace. All the computations are embarrassingly parallel. Additionally, for symmetric linear system we introduce new techniques for symmetrizing preconditioners. Many numerical examples, e.g. from PDE applications such as domain decomposition and Stokes problem, show that these new preconditioners often lead to faster convergence and smaller condition numbers. [ABSTRACT FROM AUTHOR]

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]