Showing papers on "Matrix analysis published in 2003"
TL;DR: A short introduction to methods for the data-sparse approximation of matrices resulting from the discretisation of non-local operators occurring in boundary integral methods, as the inverses of partial differential operators or as solutions of control problems.
Abstract: We give a short introduction to methods for the data-sparse approximation of matrices resulting from the discretisation of non-local operators occurring in boundary integral methods, as the inverses of partial differential operators or as solutions of control problems. The result of the approximation will be so-called hierarchical matrices (or short H-matrices). These matrices form a subset of the set of all matrices and have a data-sparse representation. The essential operations for these matrices (matrix-vector and matrix – matrix multiplication, addition and inversion) can be performed in, up to logarithmic factors, optimal complexity. We give a review of specialised variants of H-matrices, especially of H 2 -matrices, and finally consider applications of the different methods to problems from integral equations, partial differential equations and control theory. q 2003 Elsevier Science Ltd. All rights reserved.
447 citations
Book•
15 Apr 2003
TL;DR: In this article, the PSD Completion Problem Complete Positivity: Definition and Basic Properties Cones of Completely Positive Matrices Small Matrices complete positive matrix Small matrices complete positivity and the comparison matrix Completely positive graphs complete positive graph matrix complete positive graphs complete positive matrices of a given rank Complete positive matrix of the graph.
Abstract: Matrix Theoretic Background Positive Semidefinite Matrices Nonnegative Matrices and M-Matrices Schur Complements Graphs Convex Cones The PSD Completion Problem Complete Positivity: Definition and Basic Properties Cones of Completely Positive Matrices Small Matrices Complete Positivity and the Comparison Matrix Completely Positive Graphs Completely Positive Matrices Whose Graphs are Not Completely Positive Square Factorizations Functions of Completely Positive Matrices The CP Completion Problem CP Rank: Definition and Basic Results Completely Positive Matrices of a Given Rank Completely Positive Matrices of a Given Order When is the CP-Rank Equal to the Rank?
336 citations
203 citations
TL;DR: This work investigates a new approach to exploit the ℋ-matrix structure for the solution of large scale Lyapunov and Riccati equations as they typically arise for optimal control problems where the constraint is a partial differential equation of elliptic type.
Abstract: In previous papers, a class of hierarchical matrices (H-matrices) is introduced which are data-sparse and allow an approximate matrix arithmetic of almost optimal complexity. Here, we investigate a new approach to exploit the H-matrix structure for the solution of large scale Lyapunov and Riccati equations as they typically arise for optimal control problems where the constraint is a partial differential equation of elliptic type. This approach leads to an algorithm of linear-logarithmic complexity in the size of the matrices.
117 citations
TL;DR: An application to trajectory planning is presented, showing the usefulness of the present characterization of transformations expressing the system variables in terms of a linear flat output and derivatives as the kernel of a polynomial matrix.
114 citations
TL;DR: This work defines and explores the properties of the exchange operator, which maps J-orthogonal matrices to orthogonalMatrices and vice versa, and shows how the exchange operators can be used to obtain a hyperbolic CS decomposition of a J- Orthogonal matrix directly from the usual CS decompositions of an orthogsonal matrix.
Abstract: A real, square matrix Q is J-orthogonal if QTJQ = J, where the signature matrix $J = \diag(\pm 1)$. J-orthogonal matrices arise in the analysis and numerical solution of various matrix problems involving indefinite inner products, including, in particular, the downdating of Cholesky factorizations. We present techniques and tools useful in the analysis, application, and construction of these matrices, giving a self-contained treatment that provides new insights. First, we define and explore the properties of the exchange operator, which maps J-orthogonal matrices to orthogonal matrices and vice versa. Then we show how the exchange operator can be used to obtain a hyperbolic CS decomposition of a J-orthogonal matrix directly from the usual CS decomposition of an orthogonal matrix. We employ the decomposition to derive an algorithm for constructing random J-orthogonal matrices with specified norm and condition number. We also give a short proof of the fact that J-orthogonal matrices are optimally scaled und...
107 citations
TL;DR: Upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial (or λ-matrix) based on norms of the coefficient matrices and involve the inverses of the leading and trailing coefficientMatrices.
104 citations
TL;DR: RSparseM provides some basic R functionality for linear algebra with sparse matrices and a family of linear model fitting functions that implement least squares methods for problems with sparse design matrices.
Abstract: SparseM provides some basic R functionality for linear algebra with sparse matrices. Use of the package is illustrated by a family of linear model fitting functions that implement least squares methods for problems with sparse design matrices. Significant performance improvements in memory utilization and computational speed are possible for applications involving large sparse matrices.
78 citations
TL;DR: Algorithms for the solution of linear systems of equations, where the coefficient matrix can be written in the form of a banded plus semiseparable matrix, based on novel matrix factorizations developed specifically for matrices with such structures.
Abstract: We present fast and numerically stable algorithms for the solution of linear systems of equations, where the coefficient matrix can be written in the form of a banded plus semiseparable matrix. Such matrices include banded matrices, banded bordered matrices, semiseparable matrices, and block-diagonal plus semiseparable matrices as special cases. Our algorithms are based on novel matrix factorizations developed specifically for matrices with such structures. We also present interesting numerical results with these algorithms.
59 citations
TL;DR: In this article, the authors considered meet matrices on meet-semilattices as an abstract generalization of greatest common divisor (GCD) matrices, and showed that a semi-multiplicative function can be used to define the determinant of join matrices.
54 citations
TL;DR: In this paper, a general condition for the stability of a convex hull of matrices is given, which can be used to study the stability properties of interval dynamical systems.
Abstract: Using some properties of the matrix measure, we obtain a general condition for the stability of a convex hull of matrices that will be applied to study the stability of interval dynamical systems. Some classical results from stability theory are reproduced and extended. We present a relationship between the matrix measure and the real parts of the eigenvalues that make it possible to obtain stability criteria.
TL;DR: In this paper, the relation between positivity of principal minors, sign symmetry and stability of matrices is studied, and it is shown that having positive principal minors is equivalent to stability, to D-stability, and to having a positive scaling into a stable matrix.
TL;DR: In this article, two sets of new characterizations for normal matrices and EP matrices are presented, which are derived through ranks and generalized inverses of matrices, respectively.
TL;DR: In this article, the authors considered the inverse spectrum problem for nonnegative matrices and derived necessary and sufficient conditions for the existence of an n × n nonnegative matrix A with spectrum σ.
TL;DR: This contribution is a natural follow-up of the paper of the same authors entitled Convergence theory of an aggregation/disaggregation methods for computing stationary probability vectors of stochastic matrices published in [Numer. Linear Algebra Appl. 5 (1998) 253].
TL;DR: In this article, diagonal linear inequalities for three and four-body reduced density matrices were derived, which gave a different insight into the previous derivations of linear inequalities of the two-body reduction matrix.
Abstract: Diagonal linear inequalities are derived for three- and four-body reduced density matrices. These give a different insight into the previous derivations of linear inequalities for the two-body reduced density matrix. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
TL;DR: In this paper, a general block-tridiagonal matrix and the corresponding transfer matrix are related by a spectral duality by allowing for a complex Bloch parameter in the boundary conditions, and the counting function of exponents is related to winding numbers of eigenvalues.
Abstract: I consider a general block-tridiagonal matrix and the corresponding transfer matrix. By allowing for a complex Bloch parameter in the boundary conditions, the two matrices are related by a spectral duality. As a consequence, I derive some analytic properties of the exponents of the transfer matrix in terms of the eigenvalues of the (non-Hermitian) block matrix. Some of them are the single-matrix analogues of results holding for Lyapunov exponents of an ensemble of block matrices, which occur in models of transport. The counting function of exponents is related to winding numbers of eigenvalues. I discuss some implications of duality for the distribution (real bands and complex arcs) and the dynamics of eigenvalues.
Dissertation•
01 Jan 2003
TL;DR: A fraction-free algorithm for row reduction for matrices of Ore polynomials is obtained by formulating row reduction as a linear algebra problem and this algorithm is used as a basis to formulate modular algorithms for computing a row-reduced form, a weak Popov forms, and the Popov form of a polynomial matrix.
Abstract: In this thesis we study algorithms for computing normal forms for matrices of Ore polynomials while controlling coefficient growth. By formulating row reduction as a linear algebra problem, we obtain a fraction-free algorithm for row reduction for matrices of Ore polynomials. The algorithm allows us to compute the rank and a basis of the left nullspace of the input matrix. When the input is restricted to matrices of shift polynomials and ordinary polynomials, we obtain fraction-free algorithms for computing row-reduced forms and weak Popov forms. These algorithms can be used to compute a greatest common right divisor and a least common left multiple of such matrices. Our fraction-free row reduction algorithm can be viewed as a generalization of subresultant algorithms. The linear algebra formulation allows us to obtain bounds on the size of the intermediate results and to analyze the complexity of our algorithms.
We then make use of the fraction-free algorithm as a basis to formulate modular algorithms for computing a row-reduced form, a weak Popov form, and the Popov form of a polynomial matrix. By examining the linear algebra formulation, we develop criteria for detecting unlucky homomorphisms and determining the number of homomorphic images required.
17 Jan 2003
TL;DR: Numerical results that show that the coefficient matrices resulting from global spectral discretizations of certain integral equations indeed have this matrix structure are given.
Abstract: We define the class of sequentially semi-separable matrices in this paper. Essentially this is the class of matrices which have low numerical rank on their off diagonal blocks. Examples include banded matrices, semi-separable matrices, their sums as well as inverses of these sums. Fast and stable algorithms for solving linear systems of equations involving such matrices and computing Moore-Penrose inverses are presented. Supporting numerical results are also presented. In addition, fast algorithms to construct and update this matrix structure for any given matrix are presented. Finally, numerical results that show that the coefficient matrices resulting from global spectral discretizations of certain integral equations indeed have this matrix structure are given.
TL;DR: In this article, the authors describe multiplicative maps on complex and real matrices that leave invariant a certain function, property, or set of matrices: norms, spectrum, spectral radius, elementary symmetric functions of eigenvalues, certain functions of singular values, (p, q )n umerical ranges and radii, sets of unitary, normal, or Hermitian matrices, as well as sets of Hermitians with fixed inertia.
Abstract: Descriptions are given of multiplicative maps on complex and real matrices that leave invariant a certain function, property, or set of matrices: norms, spectrum, spectral radius, elementary symmetric functions of eigenvalues, certain functions of singular values, (p, q )n umerical ranges and radii, sets of unitary, normal, or Hermitian matrices, as well as sets of Hermitian matrices with fixed inertia. The treatment of all these cases is unified, and is based on general group theoretic results concerning multiplicative maps of general and special linear groups, which in turn are based on classical results by Borel - Tits. Multiplicative maps that leave invariant elementary symmetric functions of eigenvalues and spectra are described also for matrices over a general commutative field.
TL;DR: In this paper, the linear operators from the linear space of n × n matrices into the space of m × m matrices over any field F that preserve adjoint matrix, where n ⩾3.
TL;DR: In this paper, the determinant of the sum of two matrices in which both matrices are separated in the sense that the resulting expression consists of a sum of traces of products of their compound matrices.
Abstract: In this paper we demonstrate the capabilities of geometric algebra by the derivation of a formula for the determinant of the sum of two matrices in which both matrices are separated in the sense that the resulting expression consists of a sum of traces of products of their compound matrices. For the derivation we introduce a vector of Grassmann elements associated with an arbitrary square matrix, we recall the concept of compound matrices and summarize some of their properties. This paper introduces a new derivation and interpretation of the relationship between p-forms and the pth compound matrix, and it demonstrates the use of geometric algebra, which has the potential to be applied to a wide range of problems.
TL;DR: A method based on characteristic polynomial of matrix is developed to find all realroot matrices that are functions of the original 3×3 matrix, including all possible (function) stochastic root matrices.
Abstract: In this paper, we study the stochastic root matrices of stochastic matrices. All stochastic roots of 2×2 stochastic matrices are found explicitly. A method based on characteristic polynomial of matrix is developed to find all real root matrices that are functions of the original 3×3 matrix, including all possible (function) stochastic root matrices. In addition, we comment on some numerical methods for computing stochastic root matrices of stochastic matrices.
TL;DR: In this article, a set of fast real transforms including the well known Hartley transform is fully investigated and the mixed radix splitting properties of Hartley-type transforms are examined in detail.
TL;DR: In this article, the scattering and transmission of waves through a two-dimensional photonic Fabry-Perot resonator are analyzed and studied using scattering matrix theory, assuming normal incidence, single mode propagation, and sufficient inter-element spacing in the direction of propagation.
TL;DR: The general formula of the variable weighted functions w i ( t ) in the VWCF problems is obtained and the optimal weighted matrix is put forward to get the optimal weights by minimizing errors square sum J at any given times.
Abstract: In this paper, we describe a mathematical framework to determine the weighted functions in variable weight combined forecasting (VWCF) problems with continuous variable weights. Due to the polynomial approximation theorem and matrix analysis, the general formula of the variable weighted functions wi(t) in the VWCF problems is obtained. We put forward the optimal weighted matrix and get the optimal weights by minimizing errors square sum J at any given times.
TL;DR: A modified partitioning and admissibility condition is derived that ensures good convergence also for the singularly perturbed case of L∞-coefficients in the case of increasing convection.
Abstract: Hierarchical matrices provide a technique for the sparse approximation and matrix arithmetic of large, fully populated matrices. This technique has been proven to be applicable to matrices arising in the boundary and finite element method for uniformly elliptic operators with L∞-coefficients. This paper analyses the application of hierarchical matrices to the convection-dominant convection-diffusion equation with constant convection. In the case of increasing convection, the convergence of a standard H-matrix approximant towards the original matrix will deteriorate. We derive a modified partitioning and admissibility condition that ensures good convergence also for the singularly perturbed case.
TL;DR: In this paper, the authors characterized those C 2 Hn such that every matrix in the convex hull of U(C) can be written as the average of two matrices inU(C).
Abstract: LetHn be the real linear space of n n complex Hermitian matrices. The unitary (similarity) orbitU(C) of C 2 Hn is the collection of all matrices unitarily similar to C. We characterize those C 2 Hn such that every matrix in the convex hull ofU(C) can be written as the average of two matrices inU(C). The result is used to study spectral properties of submatrices of matrices inU(C), the convexity of images ofU(C) under linear transformations, and some related questions concerning the joint C-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed.
TL;DR: This chapter describes Lagrange multipliers and some selected subtopics from matrix analysis from a machine learning perspective with a detailed description of a number of mathematical constructions widely used in applied machine learning.
Abstract: This chapter describes Lagrange multipliers and some selected subtopics from matrix analysis from a machine learning perspective. The goal is to give a detailed description of a number of mathematical constructions that are widely used in applied machine learning.
TL;DR: In this paper, the authors present an inertia result for Stein equations with an indefinite right hand side, which is applied to establish connnections between the inertia of invertible hermitian block Toeplitz matrices and associated orthogonal polynomials.
Abstract: In this paper we present an inertia result for Stein equations with an indefinite right hand side. This result is applied to establish connnections between the inertia of invertible hermitian block Toeplitz matrices and associated orthogonal polynomials.