Showing papers on "Square matrix published in 2000"
TL;DR: A transformation-free form of this method that exploits incomplete Denman--Beavers square root iterations and aims for a specified accuracy (ignoring roundoff) is presented.
Abstract: The standard inverse scaling and squaring algorithm for computing the matrix logarithm begins by transforming the matrix to Schur triangular form in order to facilitate subsequent matrix square root and Pade approximation computations. A transformation-free form of this method that exploits incomplete Denman--Beavers square root iterations and aims for a specified accuracy (ignoring roundoff) is presented. The error introduced by using approximate square roots is accounted for by a novel splitting lemma for logarithms of matrix products. The number of square root stages and the degree of the final Pade approximation are chosen to minimize the computational work. This new method is attractive for high-performance computation since it uses only the basic building blocks of matrix multiplication, LU factorization and matrix inversion.
135 citations
TL;DR: Improved formulae for computing the rotations are given and it is proved that the resulting algorithm is numerically stable and can be used in certain existing Fortran implementations.
Abstract: Correlation matrices—symmetric positive semidefinite matrices with unit diagonal—are important in statistics and in numerical linear algebra. For simulation and testing it is desirable to be able to generate random correlation matrices with specified eigenvalues (which must be nonnegative and sum to the dimension of the matrix). A popular algorithm of Bendel and Mickey takes a matrix having the specified eigenvalues and uses a finite sequence of Givens rotations to introduce 1s on the diagonal. We give improved formulae for computing the rotations and prove that the resulting algorithm is numerically stable. We show by example that the formulae originally proposed, which are used in certain existing Fortran implementations, can lead to serious instability. We also show how to modify the algorithm to generate a rectangular matrix with columns of unit 2-norm. Such a matrix represents a correlation matrix in factored form, which can be preferable to representing the matrix itself, for example when the correlation matrix is nearly singular to working precision.
119 citations
TL;DR: A Lanczos-type algorithm that extends the classical Lanczos process for single starting vectors to mul- tiple starting vectors and can handle the most general case of right and left start- ing blocks of arbitrary sizes.
Abstract: Given a square matrix and single right and left starting vectors, the classical nonsymmetric Lanczos process generates two sequences of bior- thogonal basis vectors for the right and left Krylov subspaces induced by the given matrix and vectors. In this paper, we propose a Lanczos-type algorithm that extends the classical Lanczos process for single starting vectors to mul- tiple starting vectors. Given a square matrix and two blocks of right and left starting vectors, the algorithm generates two sequences of biorthogonal basis vectors for the right and left block Krylov subspaces induced by the given data. The algorithm can handle the most general case of right and left start- ing blocks of arbitrary sizes, while all previously proposed extensions of the Lanczos process are restricted to right and left starting blocks of identical sizes. Other features of our algorithm include a built-in deation procedure to detect and delete linearly dependent vectors in the block Krylov sequences, and the option to employ look-ahead to remedy the potential breakdowns that may occur in nonsymmetric Lanczos-type methods.
114 citations
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TL;DR: In this paper, the authors studied the symmetries of the square matrices and found that for six of the symmetry classes there exist simple product formulas for the number of alternating sign matrices in the class.
Abstract: An alternating sign matrix is a square matrix satisfying (i) all entries are equal to 1, -1 or 0; (ii) every row and column has sum 1; (iii) in every row and column the non-zero entries alternate in sign. The 8-element group of symmetries of the square acts in an obvious way on square matrices. For any subgroup of the group of symmetries of the square we may consider the subset of matrices invariant under elements of this subgroup. There are 8 conjugacy classes of these subgroups giving rise to 8 symmetry classes of matrices. R. P. Stanley suggested the study of those alternating sign matrices in each of these symmetry classes. We have found evidence suggesting that for six of the symmetry classes there exist simple product formulas for the number of alternating sign matrices in the class. Moreover the factorizations of certain of their generating functions point to rather startling connections between several of the symmetry classes and cyclically symmetric plane partitions.
98 citations
TL;DR: In this article, a characterization of all characteristic cones, based on the supports of the greatest common divisors (g.c.d.) of any matrix involved in the behavior description, is given.
Abstract: In the paper, the notions of characteristic set and, in particular, of characteristic cone of a two-dimensional (2-D) behavior are introduced. Autonomous behaviors are (linear shift-invariant) complete 2-D behaviors endowed with nontrivial characteristic sets. For this class of behaviors, a characterization of all characteristic cones, based on the supports of the greatest common divisors (g.c.d.'s) of the maximal order minors of any matrix involved in the behavior description, is given. Stability property of an autonomous behavior, with respect to any of its characteristic cones, is defined first for finite-dimensional behaviors and then for autonomous behaviors which are kernels of nonsingular square matrices. For both classes, stability is related to the algebraic varieties of the Laurent polynomial matrices appearing in the behavior representations. Finally, upon explicitly proving that any autonomous behavior can be expressed as the sum of a finite-dimensional behavior and of a square autonomous one, stability of general 2-D autonomous behaviors is stated and characterized.
68 citations
12 Jan 2000
TL;DR: A new proof of the well-known Lyapunov's matrix inequality about the location of the eigenvalues of a matrix in some region of the complex plane is described.
Abstract: We describe a new proof of the well-known Lyapunov's matrix inequality about the location of the eigenvalues of a matrix in some region of the complex plane. The proof makes use of standard facts from quadratic and semidefinite programming. Links are established between the Lyapunov matrix, rank-one linear matrix inequalities (LMI), and the Lagrange multiplier arising in duality theory.
50 citations
TL;DR: In this paper, the Stein's theorem in discrete linear dynamical systems was shown to hold in the open unit disk of the complex plane if and only if for every Hermitian matrix Q, there exists a positive semidefinite matrix X such that X−AXA * + Q is positive semidefinite and X[X−AXa * +Q]=0.
45 citations
TL;DR: In this paper, a simple modification of the centring of the matrix, coupled with the corresponding change in row and column masses and row-and column metrics, allows the table to be decomposed into symmetric and skew symmetric components, which can then be analysed separately.
Abstract: The application of correspondence analysis to square asymmetric tables is often unsuccessful because of the strong role played by the diagonal entries of the matrix, obscuring the data off the diagonal. A simple modification of the centring of the matrix, coupled with the corresponding change in row and column masses and row and column metrics, allows the table to be decomposed into symmetric and skew symmetric components, which can then be analysed separately. The symmetric and skew symmetric analyses can be performed by using a simple correspondence analysis program if the data are set up in a special block format. The methodology is demonstrated on a social mobility table from the first democratically elected Parliament in Germany, the Frankfurter Nationalversammlung. The table cross-tabulates the jobs of parliamentarians when first entering the labour market and their jobs in May 1848 when the Parliament started its first session.
43 citations
TL;DR: The meaning of each sequential tableau appearing during the pivoting process is interpreted and it is shown that each tableau of the process corresponds to the inverse of a row modified matrix and contains the generators of the linear subspace orthogonal to a set of vectors and its complement.
Abstract: In this paper we discuss the power of a pivoting transformation introduced by Castillo, Cobo, Jubete, and Pruneda [Orthogonal Sets and Polar Methods in Linear Algebra: Applications to Matrix Calculations, Systems of Equations and Inequalities, and Linear Programming, John Wiley, New York, 1999] and its multiple applications. The meaning of each sequential tableau appearing during the pivoting process is interpreted. It is shown that each tableau of the process corresponds to the inverse of a row modified matrix and contains the generators of the linear subspace orthogonal to a set of vectors and its complement. This transformation, which is based on the orthogonality concept, allows us to solve many problems of linear algebra, such as calculating the inverse and the determinant of a matrix, updating the inverse or the determinant of a matrix after changing a row (column), determining the rank of a matrix, determining whether or not a set of vectors is linearly independent, obtaining the intersection of two linear subspaces, solving systems of linear equations, etc. When the process is applied to inverting a matrix and calculating its determinant, not only is the inverse of the final matrix obtained, but also the inverses and the determinants of all its block main diagonal matrices, all without extra computations.
42 citations
TL;DR: In this paper, the trace of matrices is used to characterize the joint spectral radius for a bounded collection of the square matrices with complex entries and of the same size, and this characterization allows us to give some estimates concerning the computation of the joint spectrum radius.
33 citations
TL;DR: The Jordan canonical form of a matrix is used to derive systematically the Drazin inverse of modification of a square matrix, in addition to perturbation formula for the Dazin inverse.
TL;DR: In this paper, it was shown that for a field of characteristic ≠ 2, each square matrix with entries in K can be written as a product of circulant and diagonal matrices with entries.
TL;DR: A Lanczos process is constructed on a large and sparse matrix and the results are used to compute the inverse square root of the same matrix using the theory of chiral fermions on the lattice.
18 Jun 2000
TL;DR: A new version of RGEQRF and its accompanying SMP parallel counterpart is presented, implemented for a future release of the IBM ESSL library and represents a robust high-performance piece of library software for QR factorization on uniprocessor and multiprocessors systems.
Abstract: In [5,6], we presented algorithm RGEQR3, a purely recursive formulation of the QR factorization. Using recursion leads us to a natural way to choose the k-way aggregating Householder transform of Schreiber and Van Loan [10]. RGEQR3 is a performance critical subroutine for the main (hybrid recursive) routine RGEQRF for QR factorization of a general m × n matrix. This contribution presents a new version of RGEQRF and its accompanying SMP parallel counterpart, implemented for a future release of the IBM ESSL library. It represents a robust high-performance piece of library software for QR factorization on uniprocessor and multiprocessor systems. The implementation builds on previous results [5,6]. In particular, the new version is optimized in a number of ways to improve the performance; e.g., for small matrices and matrices with a very small number of columns. This is partly done by including mini blocking in the otherwise pure recursive RGEQR3. We describe the salient features of this implementation. Our serial implementation outperforms the corresponding LAPACK routine by 10-65% for square matrices and 10-100% on tall and thin matrices on the IBM POWER2 and POWER3 nodes. The tests covered matrix sizes which varied from very small to very large. The SMP parallel implementation shows close to perfect speedup on a 4-processor PPC604e node.
TL;DR: In this article, the exact Fisher information matrix of a multivariate Gaussian time series model expressed in state space form is derived by applying matrix differential rules for the derivatives of a matrix function J=J(θ) with respect to its vector argument.
TL;DR: This paper deals with a direct derivation of Fisher's information matrix of vector state space models for the general case, by which is meant the establishment of the matrix as a whole and not element by element.
Book•
01 Jan 2000
TL;DR: In this article, a structural approach to index of differential-algebraic equations is presented. But this approach is restricted to the case of mixed matrices, and it is not suitable for general matrices.
Abstract: Preface I. Introduction to Structural Approach --- Overview of the Book 1 Structural Approach to Index of DAE 1.1 Index of differential-algebraic equations 1.2 Graph-theoretic structural approach 1.3 An embarrassing phenomenon 2 What Is Combinatorial Structure? 2.1 Two kinds of numbers 2.2 Descriptor form rather than standard form 2.3 Dimensional analysis 3 Mathematics on Mixed Polynomial Matrices 3.1 Formal definitions 3.2 Resolution of the index problem 3.3 Block-triangular decomposition II. Matrix, Graph and Matroid 4 Matrix 4.1 Polynomial and algebraic independence 4.2 Determinant 4.3 Rank, term-rank and generic-rank 4.4 Block-triangular forms 5 Graph 5.1 Directed graph and bipartite graph 5.2 Jordan-Holder-type theorem for submodular functions 5.3 Dulmage-Mendelsohn decomposition 5.4 Maximum flow and Menger-type linking 5.5 Minimum cost flow and weighted matching 6 Matroid 6.1 From matrix to matroid 6.2 Basic concepts 6.3 Examples 6.4 Basis exchange properties 6.5 Independent matching problem 6.6 Union 6.7 Bimatroid (linking system) III. Physical Observations for Mixed Matrix Formulation 7 Mixed Matrix for Modeling Two Kinds of Numbers 7.1 Two kinds of numbers 7.2 Mixed matrix and mixed polynomial matrix 8 Algebraic Implications of Dimensional Consistency 8.1 Introductory comments 8.2 Dimensioned matrix 8.3 Total unimodularity of dimensioned matrices 9 Physical Matrix 9.1 Physical matrix 9.2 Physical matrices in a dynamical system IV. Theory and Application of Mixed Matrices 10 Mixed Matrix and Layered Mixed Matrix 11 Rank of Mixed Matrices 11.1 Rank identities for LM-matrices 11.2 Rank identities for mixed matrices 11.3 Reduction to independent matching problems 11.4 Algorithms for the rank 11.4.1 Algorithm for LM-matrices 11.4.2 Algorithm for mixed matrices 12 Structural Solvability of Systems of Equations 12.1 Formulation of structural solvability 12.2 Graphical conditions for structural solvability 12.3 Matroidal conditions for structural solvability 13. Combinatorial Canonical Form of LM-matrices 13.1 LM-equivalence 13.2 Theorem of CCF 13.3 Construction of CCF 13.4 Algorithm for CCF 13.5 Decomposition of systems of equations by CCF 13.6 Application of CCF 13.7 CCF over rings 14 Irreducibility of LM-matrices 14.1 Theorems on LM-irreducibility 14.2 Proof of the irreducibility of determinant 15 Decomposition of Mixed Matrices 15.1 LU-decomposition of invertible mixed matrices 15.2 Block-triangularization of general mixed matrices 16 Related Decompositions 16.1 Partition as a matroid union 16.2 Multilayered matrix 16.3 Electrical network with admittance expression 17 Partitioned Matrix 17.1 Definitions 17.2 Existence of proper block-triangularization 17.3 Partial order among blocks 17.4 Generic partitioned matrix 18 Principal Structures of LM-matrices 18.1 Motivations 18.2 Principal structure of submodular systems 18.3 Principal structure of generic matrices 18.4 Vertical principal structure of LM-matrices 18.5 Horizontal principal structure of LM-matrices V. Polynomial Matrix and Valuated Matroid 19 Polynomial/Rational Matrix 19.1 Polynomial matrix and Smith form 19.2 Rational matrix and Smith-McMillan form at infinity 19.3 Matrix pencil and Kronecker form 20 Valuated Matroid 20.1 Introduction 20.2 Examples 20.3 Basic operations 20.4 Greedy algorithms 20.5 Valuated bimatroid 20.6 Induction through bipartite graphs 20.7 Characterizations 20.8 Further exchange properties 20.9 Valuated independent assignment problem 20.10 Optimality criteria 20.10.1 Potential criterion 20.10.2 Negative-cycle criterion 20.10.3 Proof of the optimality criteria 20.10.4 Extension to VIAP(k) 20.11 Application to triple matrix product 20.12 Cycle-canceling algorithms 20.12.1 Algorithms 20.12.2 Validity of the minimum-ratio cycle algorithm 20.13 Augmenting algorithms 20.13.1 Algorithms 20.13.2 Validity of the augmenting algorithm VI. Theory and Application of Mixed Polynomial Matrices 21 Descriptions of Dynamical Systems 21.1 Mixed polynomial mat
TL;DR: A binary difference pattern is a pattern obtained by covering an equilateral triangular grid by black and white circles in a dense hexagonal packing under a simple symmetric local matching rule that allows to find all symmetric solutions and the cardinalities of the different symmetry classes.
TL;DR: In this article, the authors deal with the problem of establishing upper bounds for the norm of the nth power of square matrices, which is of central importance in the stability analysis of numerical methods for solving (linear) initial value problems for ODEs.
TL;DR: The period of a matrix is shown to be the least common multiple of the high periods of all non-trivial highly connected components in the corresponding digraph of A.
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TL;DR: The notion of the symmetrized determinant was introduced in this paper for a fixed finite-dimensional associative algebra, and it was shown that for any fixed-dimensional algebra, the determinant of a matrix with entries in the algebra can be computed in polynomial time.
Abstract: We introduce a new notion of the determinant, called symmetrized determinant, for a square matrix with the entries in an associative algebra $\AA$. The monomial expansion of the symmetrized determinant is obtained from the standard expansion of the commutative determinant by averaging the products of entries of the matrix in all possible orders. We show that for any fixed finite-dimensional associative algebra $\AA$, the symmetrized determinant of an $n\times n$ matrix with the entries in $\AA$ can be computed in polynomial in $n$ time (the degree of the polynomial is linear in the dimension of $\AA$). Then, for every associative algebra $\AA$ endowed with a scalar product and unbiased probability measure, we construct a randomized polynomial time algorithm to estimate the permanent of non-negative matrices. We conjecture that if $\AA=\Mat(d, {\Bbb R})$ is the algebra of $d\times d$ real matrices endowed with the standard scalar product and Gaussian measure, the algorithm approximates the permanent of a non-negative $n \times n$ matrix within $O(\gamma_d^n)$ factor, where $\lim_{d \longrightarrow +\infty} \gamma_d=1$. Finally, we provide some informal arguments why the conjecture might be true.
TL;DR: In this article, absolute and relative perturbation bounds are derived for angles between invariant subspaces of complex square matrices, in the two-norm and in the Frobenius norm.
TL;DR: Combinatorial and probabilistic arguments are used to prove that if the order of a ray-nonsingular matrix is at least 6, then it must contain a zero entry, and that if each of its rows and columns have an equal number, k, of nonzeros, then k613.
TL;DR: In this paper, the authors studied the N-extremal matrices of measures associated to a completely indeterminate matrix moment problem, i.e., those matrices for which the linear space of matrix polynomials is dense in the corresponding L2(W) measure.
TL;DR: It is shown, through a unified analysis, that analogous decision problems for the other matrix classes are also co-NP-complete.
Abstract: The classes of P-, P0-, R0-, semimonotone, strictly semimonotone, column sufficient, and nondegenerate matrices play important roles in studying solution properties of equations and complementarity problems and convergence/complexity analysis of methods for solving these problems. It is known that the problem of deciding whether a square matrix with integer/rational entries is a P- (or nondegenerate) matrix is co-NP-complete. We show, through a unified analysis, that analogous decision problems for the other matrix classes are also co-NP-complete.
06 Sep 2000
TL;DR: In this paper, an extension of the Parikh mapping is introduced, based on square matrices of a certain form, and the classical Parikh vector appears in such a matrix as the second diagonal.
Abstract: In this paper we introduce an extension of the Parikh mapping and investigate some of its basic properties. The extension is based on square matrices of a certain form. The classical Parikh vector appears in such a matrix as the second diagonal. However, the matrix product gives more information about a word than the Parikh vector. We characterize the matrix products and establish also an interesting interconnection between mirror images of words and inverses of matrices.
TL;DR: In this article, the Hermitian property of the matrices under study contributes to construct the link between the two approaches, and the theorem is further illustrated by an example, which shows that a considerable reduction in matrix integrals is taking place when moving from the former to the latter.
TL;DR: A method for extracting information carrying eigenvalues of the correlation matrix is presented based on the topological transformation of the manifold defined by the data matrix itself, showing that the results are superior to the results of the random matrix theory as applied to the financial data.
Abstract: A method for extracting information carrying eigenvalues of the correlation matrix is presented based on the topological transformation of the manifold defined by the data matrix itself. The transformation, performed with the use of the minimum spanning tree and the barycentric transformation, linearizes the topological manifold and the singular value decomposition is performed on the final data matrix corresponding to the linearized hypersurface. It is shown that the results of this procedure are superior to the results of the random matrix theory as applied to the financial data. The method may be used independently or in conjunction with the random matrix theory. Other possible uses of the method are mentioned.
TL;DR: New trace bounds for the product of two general matrices are proposed by replacing eigenvalues partly by singular values in the equation of bounds to remove the restriction of symmetry.
Abstract: Estimates of bounds on the solutions of Lyapunov and Riccati equations are important for analysis and synthesis of linear systems. In this paper, we propose new trace bounds for the product of two general matrices. The key point for removing the restriction of symmetry is to replace eigenvalues partly by singular values in the equation of bounds. The results obtained are valid for both symmetric and nonsymmetric cases and give tighter bounds in certain cases.