Embedding rectangular matrices in hadamard matrices

It is shown that any m×n±1 matrix may be embedded in a Hadamard matrix of order kl, where k and l are the least orders greater than or equal to m and nrespectively in which Hadamard matrices exist.     

The CMP inverse for rectangular matrices

We extend the notation of the CMP inverse for a square matrix to a rectangular matrix. Precisely, we define and characterize a new generalized inverse called the weighted CMP inverse. Also, we investigate properties of the weighted CMP inverse using a representation by block matrices. Some new characterizations and properties of the CMP inverse are obtained.     

A Stein-Rosenberg theorem for rectangular matrices

In the theory of iterative methods, the classical Stein-Rosenberg theorem can be viewed as giving the simultaneous convergence (or divergence) of the extrapolated Jacobi (JOR) matrix $\text{J}$_ω and the successive overrelaxation (SOR) matrix $\text{L}$_ω, in the case when the Jacobi matrix $\text{J}$₁ is nonegative. As has been established by Buoni and Varga, necessary and sufficient conditions for the simultaneous convergence (or divergence) of $\text{J}$_ω and $\text{L}$_ω have been established which do not depend on the assumption that $\text{J}$₁ is nonnegative. More recently, Buoni, Neumann, and Varga extended these results to the singular case, using the notion of semiconvergence. The aim here is to extend these results to consistent rectangular systems.     

A Drazin inverse for rectangular matrices

The definition of the Drazin inverse of a square matrix with complex elements is extended to rectangular matrices by showing that for any B and W,m by n and n by m, respectively, there exists a unique matrix, X, such that (BW)^k=(BW)^k+1XW for some positive integer k, XWBWX = X, and BWX = XWB. Various expressions satisfied by B, W,X and related matrices are developed.     

Pseudospectra of rectangular matrices

Pseudospectra of rectangular matrices vary continuously withthe matrix entries, a feature that eigenvalues of these matricesdo not have. Some properties of eigenvalues and pseudospectraof rectangular matrices are explored, and an efficient algorithmfor the computation of pseudospectra is proposed. Applicationsare given in (square) eigenvalue computation (Lanczos iteration),square pseudospectra approximation (Arnoldi iteration), controltheory (nearest uncontrollable system) and game theory.     

The Binet-Pexider functional equation for rectangular matrices

IfK is a field of characteristic 0 then the following is shown. Iff, g, h: M _n (K) K are non-constant solutions of the Binet—Pexider functional equation

A convergent splitting of matrices

LetA, M, N ben ×n real matrices, letA = M– N, letA andM be nonsingular, letM y 0 implyN y 0, and letA y 0 implyN y 0 (where the prime denotes the transpose). Then the spectral radius(M ^–1 N) ofM ^–1 N is less than one, and the iterative processx ⁱ⁺¹ =M ^–1 N x ⁱ +M ^–1 b converges to the solution ofA x = b starting from anyx ⁰.Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No. DA-31-124-ARO-D-462, and in part by the National Science Foundation under Grant NSF GP-6070.     

Graph properties for splitting with grounded Laplacian matrices

With any undirected, connected graphG containing no self-loops one can associate the Laplacian matrixL(G). It is also the (singular) admittance matrix of a resistive network with all conductances taken to be unity. While solving the linear system involved, one of the vertices is grounded, so the coefficient matrix is a principal submatrix ofL which we will call the grounded Laplacian matrixL ₁. In this paper we consider iterative solutions of such linear systems using certain regular splittings ofL ₁ and derive an upper bound for the spectral radius of the iteration matrix in terms of the properties of the graphG.This work was supported by the Academy of Finland     

Simultaneous diagonalization of rectangular matrices

A matrix D is said to be diagonal if its (i,j)th element is null whenever i and j are unequal. For a set {A_θ} of matrices A_θ of the same order, the paper gives necessary and sufficient conditions for nonsingular matrices S and T to exist, such that SA_θT = D_θ is diagonal for each matrix A_θ in the set.     

On eigenvalues of rectangular matrices

Given a (k+1)-tuple A,B ₁, ..., B _k of m×n matrices with m ≤ n, we call the set of all k-tuples of complex numbers {λ ₁, ..., λ k} such that the linear combination A+λ ₁ B ₁+λ ₂ B ₂+ ... +λ _k B _k has rank smaller than m the eigenvalue locus of the latter pencil. Motivated primarily by applications to multiparameter generalizations of the Heine-Stieltjes spectral problem, we study a number of properties of the eigenvalue locus in the most important case k = n−m+1.     

Inverting graphs of rectangular matrices

This paper addresses the question of determining the class of rectangular matrices having a given graph as a row or column graph. We also determine equivalent conditions on a given pair of graphs in order for them to be the row and column graphs of some rectangular matrix. In connection with these graph inversion problems we discuss the concept of minimal inverses. This concept turns out to have two different forms in the case of one-graph inversion. For the two-graph case we present a method of determining when an inverse is minimal. Finally we apply the two-graph theorem to a class of energy related matrices.     

A block Cholesky-LU-based QR factorization for rectangular matrices

The Householder method provides a stable algorithm to compute the full QR factorization of a general matrix. The standard version of the algorithm uses a sequence of orthogonal reflections to transform the matrix into upper triangular form column by column. In order to exploit (level 3 BLAS or structured matrix) computational advantages for block-partitioned algorithms, we develop a block algorithm for the QR factorization. It is based on a well-known block version of the Householder method which recursively divides a matrix columnwise into two smaller matrices. However, instead of continuing the recursion down to single matrix columns, we introduce a novel way to compute the QR factors in implicit Householder representation for a larger block of several matrix columns, that is, we start the recursion at a block level instead of a single column. Numerical experiments illustrate to what extent the novel approach trades some of the stability of Householder's method for the computational efficiency of block methods.     

Further study of alternating iterations for rectangular matrices

Theory of matrix splittings is a useful tool in the analysis of iterative methods for solving systems of linear equations. When two splittings are given, it is of interest to compare the spectral radii of the corresponding iteration matrices. This helps to arrive at the conclusion that which splitting should one choose so that one can reach the desired solution of accuracy or the exact solution in a faster way. In the case of many splittings are provided, the comparison of the spectral radii is time-consuming. Such a situation can be overcome by introducing another iteration scheme which converges to the same solution of interest in a much faster way. In this direction, the theory of alternating iterations for real rectangular matrices is recently proposed. In this note, some more results to the theory of alternating iterations are added. A comparison result of two different alternating iteration schemes is then presented which will help us to choose the iteration scheme that will guarantee the faster convergence of the alternating iteration scheme. In addition to these results, a comparison result for proper weak regular splittings is also obtained.     

Optimal in-place transposition of rectangular matrices

Given a rectangular m×n matrix stored as a two-dimensional array, we want to transpose it in place and measure the cost by the number of memory writes and the number of auxiliary cells used. We propose a transposition algorithm with optimal complexity O(mn) using only min(m,n) auxiliary memory cells.     

Certain isometries of rectangular complex matrices

If s₁(A) ? ? ? s_m(A) are the singular values of A ? M_m,n(C), and if 1 ?k ?m ? and p ? 1, then

$\text{φ}{}_{\mathrm{p,k}}\text{(A) = (}\text{∑}\text{i=1}\text{k}\text{s}{}_{\mathrm{i}}{}^{\mathrm{p}}\text{(A)}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}$

is a unitarily invariant norm. In this paper a complete determination of the extreme points on the corresponding unit spheres is accomplished in all cases, enabling the isometries with respect to Φ_p,_k to be determined in the case p = 1. This removes the restriction m = n in an earlier paper of the author and Marcus.     

Geometry of rectangular block triangular matrices

Let D be any division ring, and let T（mi,ni,k） be the set of k × k （k ≥ 2） rectangular block triangular matrices over D. For A, B ∈ T（mi,ni,k）, if rank（A - B） = 1, then A and B are said to be adjacent and denoted by A -B. A map T ： T（mi,ni,k） -〉 T（mi,ni,k） is said to be an adjacency preserving map in both directions if A - B if and only if φ（A） φ（B）. Let G be the transformation group of all adjacency preserving bijections in both directions on T（mi,ni,k）. When m1,nk ≥ 2, we characterize the algebraic structure of G, and obtain the fundamental theorem of rectangular block triangular matrices over D.     

Semiconvexity of invariant functions of rectangular matrices

The paper deals with the semiconvexity properties (i.e., the rank 1 convexity, quasiconvexity, polyconvexity, and convexity) of rotationally invariant functions f of matrices. For the invariance with respect to the proper orthogonal group and the invariance with respect to the full orthogonal group coincide. With each invariant f one can associate a fully invariant function of a square matrix of type where It is shown that f has the semi convexity of a given type if and only if has the semiconvexity of that type. Consequently the semiconvex hulls of f can be determined by evaluating the corresponding hulls of and then extending them to matrices by rotational invariance. Received: 10 October 2001 / Accepted: 23 January 2002 // Published online: 6 August 2002 RID="*" ID="*" This research was supported by Grant 201/00/1516 of the Czech Republic.     

Natural density of rectangular unimodular integer matrices

In this paper, we compute the natural density of the set of k×n integer matrices that can be extended to an invertible n×n matrix over the integers. As a corollary, we find the density of rectangular matrices with Hermite normal form . Connections with Cesàro’s Theorem on the density of coprime integers and Quillen-Suslin’s Theorem are also presented.     

Spectral multiplicity and splitting results for a class of qualitative matrices

This paper presents some results concerning the location and multiplicity of eigenvalues of sign symmetric matrices whose associated graphs are trees. In particular it extends previous spectral multiplicity and splitting results proved by others.