Numerical exterior algebra and the compound matrix method

Summary. The compound matrix method, which was first proposed for numerically integrating systems of differential equations in hydrodynamic stability on k=2,3 dimensional subspaces of , by using compound matrices as coordinates, is reformulated in a coordinate-free way using exterior algebra spaces, . This formulation leads to a general framework for studying systems of differential equations on k-dimensional subspaces. The framework requires the development of several new ideas: the role of Hodge duality and the Hodge star operator in the construction, an efficient strategy for constructing the induced differential equations on , general formulation of induced boundary conditions, the role of geometric integrators for preserving the manifold of k-dimensional subspaces – the Grassmann manifold, , and a formulation for induced systems on an unbounded interval. The numerical exterior algebra framework is most advantageous for numerical solution of differential eigenvalue problems on unbounded domains, where there are significant difficulties in setting up matrix discretizations. The formulation is presented for k-dimensional subspaces of systems on with k and n arbitrary, and examples are given for the cases of k=2 and n=4, and k=3 and n=6, with an indication of implementation details for systems of larger dimension. The theory is illustrated by application to four differential eigenvalue problems on unbounded intervals: hydrodynamic stablity of boundary-layer flow past a compliant surface, the eigenvalue problem associated with the stability of solitary waves, the stability of Bickley jet in oceanography, and the eigenvalue problem associated with the stability of the Ekman layer in atmospheric dynamics. Received February 2, 2001 / Revised version received May 28, 2001 / Published online October 17, 2001     

Generalized matrix near-rings

Let R be a right near-ring with identity. The k×k matrix near-ring over R, Mat_k(R R), as defined by Meldrum and van der Walt, regards R as a left mod-ule over R. Let M be any faithful left R-module. Using the action of R on M, a generalized k×k matrix near-ring, Mat_k(R M), is defined. It is seen that Mat_k(R M) has many of the features of Mat_k(R R). Differences be-tween the two classes of near-rings are shown. In spe- cial cases there are relationships between Mat_k(R M) and Mat_k(R R). Generalized matrix near-rings Mat_k(R M) arise as the “right near-ring” of finite centraiizer near-rings of the form M _A{G)> where G is a finite group and A is a fixed point free automorphism group on G.     

Generalized matrix Pade approximants

For rectangular matrix functions with restricted sizes of their column and row, we introduce the problem of Padé approximation similar to its scalar counterpart. Results on the existence and uniqueness of the approximants are given. Determinantal expressions and some properties of the approximants are established. Supported by Science Fund for Youth of Chinese Academy of Sciences.     

Generalized matrix functions and determinants

In this paper we prove that, up to a scalar multiple, the determinant is the unique generalized matrix function that preserves the product or remains invariant under similarity. Also, we present a new proof for the known result that, up to a scalar multiple, the ordinary characteristic polynomial is the unique generalized characteristic polynomial for which the Cayley-Hamilton theorem remains true.     

Generalized matrix inversion is not harder than matrix multiplication

Starting from the Strassen method for rapid matrix multiplication and inversion as well as from the recursive Cholesky factorization algorithm, we introduced a completely block recursive algorithm for generalized Cholesky factorization of a given symmetric, positive semi-definite matrix A∈R^n×n$A\in {\mathbb{R}}^{n\times n}$. We used the Strassen method for matrix inversion together with the recursive generalized Cholesky factorization method, and established an algorithm for computing generalized {2,3}$\{2,3\}$ and {2,4}$\{2,4\}$ inverses. Introduced algorithms are not harder than the matrix–matrix multiplication.     

Generalized compound quadrature formulae for finite-part integrals

Received on 31 July 1995. Revised on 19 August 1996. We investigate the error term of the dth degree compound quadratureformulae for finite-part integrals of the form where and p 1.We are mainly interested in error bounds of the form with best possible constants c. Itis shown that, for and n uniformlydistributed nodes, the error behaves as O(n^p–s–1for , p–1 <s d+1.In a previous paper we have shown that this is not true for As an improvement, we consider the case of non-uniformly distributednodes. Here, we show that for all p I and , an O(n^–s) error estimate can be obtainedin theory by a suitable choice of the nodes. A set of nodeswith this property is staled explicitly. In practice, this gradedmesh causes stability problems which are computationally expensiveto overcome. E-mail address: diethelm{at}informatik.uni-hildesheim.de     

Generalized pseudoinverses of matrix valued functions

We define the pseudoinverse (resp. a generalized pseudoinverse) of a matrix-valued functionF to be the functionF ^x such that, for each in the domain ofF, F ^x () is the inverse (resp. a generalized inverse) of the matrixF(). We derive a state space formula for a generalized pseudoinverse of a rational matrix function without a pole or zero at infinity. This derivation makes use of the theorem characterizing the factorization of a nonregular rational matrix functionW in terms of the decomposition of the state space of a realization ofW. We also give a formula for a generalized pseudoinverse of an arbitrary rational matrix function in the form of a centered realization. We indicate some applications of generalized pseudoinverses of matrix valued functions.     

Generalized state-space system matrix equivalents of a Rosenbrock system matrix

Bosgra & Van Der Weiden (1981) have given a procedure wherebya Rosenbrock system matrix may be reduced to an equivalent generalizedstate-space system matrix. The sense in which this is equivalentto the original system matrix is that the reduced system matrixexhibits identical system properties both at finite and infinitefrequencies. Hayton et al. (1990) introduced the transformationsof normal full system equivalence and full system equivalence.In the present work, we show that the Bosgra & Van Der Weidenreduction procedure is a full system-equivalence transformation,and a characterization of this equivalence in a matrix-transformationsense is also provided.     

Generalized Jordan derivations on triangular matrix algebras

In this note, we prove that every generalized Jordan derivation from the algebra of all upper triangular matrices over a commutative ring with identity into its bimodule is the sum of a generalized derivation and an antiderivation.     

Generalized Jordan matrix of a linear operator

For any linear operator defined over an arbitrary field k, there is a basis in which this matrix is a generalized Jordan matrix (of the second kind) with elements in the field k. For any linear operator, such a matrix is defined uniquely up to permutation of diagonal blocks.     

Generalized Green's matrix for linear pulse boundary-value problems

We establish an algebraic criterion of solvability, study the structure of general solutions of linear boundary-value problems for systems of differential equations with pulse effects, and construct the generalized Green's matrix.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 849–856, July, 1994.     

Generalized latin matrix and construction of orthogonal arrays

In this paper, generalized Latin matrix and orthogonal generalized Latin matrices are proposed. By using the property of orthogonal array, some methods for checking orthogonal generalized Latin matrices are presented. We study the relation between orthogonal array and orthogonal generalized Latin matrices and obtain some useful theorems for their construction. An example is given to illustrate applications of main theorems and a new class of mixed orthogonal arrays are obtained.     

Generalized Bayes minimax estimation of the normal mean matrix with unknown covariance matrix

This paper addresses the problem of estimating the normal mean matrix in the case of unknown covariance matrix. This problem is solved by considering generalized Bayesian hierarchical models. The resulting generalized Bayes estimators with respect to an invariant quadratic loss function are shown to be matricial shrinkage equivariant estimators and the conditions for their minimaxity are given.     

Generalized matrix version of reverse Hölder inequality

A generalized matrix version of reverse Cauchy-Schwarz/Hölder inequality is proved. This includes the recent results proved by Bourin, Fujii, Lee, Niezgoda and Seo.     

Generalized matrix versions of the Cauchy-Schwarz and Kantorovich inequalities

Summary A recent note by Marshall and Olkin (1990), in which the Cauchy-Schwarz and Kantorovich inequalities are considered in matrix versions expressed in terms of the Loewner partial ordering, is extended to cover positive semidefinite matrices in addition to positive definite ones.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

On minors of the compound matrix of a Laplacian

Let L   be an n×n$n\times n$ matrix with zero row and column sums, n?3$n?3$. We obtain a formula for any minor of the (n−2)$(n-2)$-th compound of L. An application to counting spanning trees extending a given forest is given.