Showing papers on "Matrix analysis published in 2001"

On the 1/n Expansion for Some Unitary Invariant Ensembles of Random Matrices

Abstract: We present a version of the 1/n-expansion for random matrix ensembles known as matrix models. The case where the support of the density of states of an ensemble consists of one interval and the case where the density of states is even and its support consists of two symmetric intervals is treated. In these cases we construct the expansion scheme for the Jacobi matrix determining a large class of expectations of symmetric functions of eigenvalues of random matrices, prove the asymptotic character of the scheme and give an explicit form of the first two terms. This allows us, in particular, to clarify certain theoretical physics results on the variance of the normalized traces of the resolvent of random matrices. We also find the asymptotic form of several related objects, such as smoothed squares of certain orthogonal polynomials, the normalized trace and the matrix elements of the resolvent of the Jacobi matrices, etc.

The generalized star product and the factorization of scattering matrices on graphs

Abstract: In this article we continue our analysis of Schrodinger operators on arbitrary graphs given as certain Laplace operators. In the present article we give the proof of the composition rule for the scattering matrices. This composition rule gives the scattering matrix of a graph as a generalized star product of the scattering matrices corresponding to its subgraphs. We perform a detailed analysis of the generalized star product for arbitrary unitary matrices. The relation to the theory of transfer matrices is also discussed.

The Spectrum of Random Matrices

Abstract: The Frobenius eigenvector of a positive square matrix is obtained by iterating the multiplication of an arbitrary positive vector by the matrix. Brody (1997) noticed that, when the entries of the matrix are independently and identically distributed, the speed of convergence increases statistically with the dimension of the matrix. As the speed depends on the ratio between the subdominant and the dominant eigenvalues, Brody's conjecture amounts to stating that this ratio tends to zero when the dimension tends to infinity. The paper provides a simple proof of the result. Some mathematical and economic aspects of the problem are discussed.

Optimal Stability and Eigenvalue Multiplicity

Abstract: We consider the problem of minimizing over an affine set of square matrices the maximum of the real parts of the eigenvalues. Such problems are prototypical in robust control and stability analysis. Under nondegeneracy conditions, we show that the multiplicities of the active eigenvalues at a critical matrix remain unchanged under small perturbations of the problem. Furthermore, each distinct active eigenvalue corresponds to a single Jordan block. This behavior is crucial for optimality conditions and numerical methods. Our techniques blend nonsmooth optimization and matrix analysis.

Matrix Algebra: Exercises and Solutions

Abstract: 1 Matrices.- 2 Submatrices and Partitioned Matrices.- 3 Linear Dependence and Independence.- 4 Linear Spaces: Rowand Column Spaces.- 5 Trace of a (Square) Matrix.- 6 Geometrical Considerations.- 7 Linear Systems : Consistency and Compatibility.- 8 Inverse Matrices.- 9 Generalized Inverses.- 10 Idempotent Matrices.- 11 Linear Systems: Solutions.- 12 Projections and Projection Matrices.- 13 Determinants.- 14 Linear, Bilinear, and Quadratic Forms.- 15 Matrix Differentiation.- 16 Kronecker Products and the Vec and Vech Operators.- 17 Intersections and Sums of Subspaces.- 18 Sums (and Differences) of Matrices.- 19 Minimization of a Second-Degree Polynomial (in n Variables) Subject to Linear Constraints.- 20 The Moore-Penrose Inverse.- 21 Eigenvalues and Eigenvectors.- 22 Linear Transformations.- References.

Spectra, Pseudospectra, and Localization for Random Bidiagonal Matrices

Abstract: There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random, non-Hermitian, periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a “bubble with wings” in the complex plane, and the corresponding eigenvectors are localized in the wings, delocalized in the bubble. Here, in addition to eigenvalues, pseudospectra are analyzed, making it possible to treat the nonperiodic analogues of these random matrix problems. Inside the bubble, the resolvent norm grows exponentially with the dimension. Outside, it grows subexponentially in a bounded region that is the spectrum of the infinite-dimensional operator. Localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase, respectively, with distance from the diagonal. This article presents theorems that characterize the spectra, pseudospectra, and numerical range for the four cases of finite bidiagonal matrices, infinite bidiagonal matrices (“stochastic Toeplitz operators”), finite periodic matrices, and doubly infinite bidiagonal matrices (“stochastic Laurent operators”). c 2001 John Wiley & Sons, Inc.

Matrix Calculus and Zero-One Matrices: Statistical and Econometric Applications

Abstract: 1. Classical statistical procedures 2. Elements of matrix algebra 3. Zero-one matrices 4. Matrix calculus 5. Linear regression models 6. Seemingly unrelated regression equations models 7. Linear simultaneous equations models.

Matrix Analysis Of Structural Dynamics: Applications And Earthquake Engineering

Abstract: Characteristics of free and forced vibrations of elementary systems Eigensolution techniques and undamped response analysis of multiple-degree-of-freedom systems Eigensolution methods and response analysis for proportional and non-proportional damping dynamic stiffness and energy methods for distributed mass systems dynamic stiffness method for coupling vibration, elastic media and Pdeltal effect consistent mass method of frames and finite elements numerical integration methods and seismic response spectra for single- and multi-components seismic input formulation and response analysis of 3-D building systems with walls and bracings various hysteresis models of non-linear response analysis static and dynamic lateral-force procedures and related effects in building codes of UBC-94, UBC-97, and IBC-2000 problems solutions. Appendices: Lagrange's equation derivation of ground rotation vector analysis fundamentals transformation matrix between JCS and GCS transformation matrix between ECS and GCS for beam column transformation matrix and stiffness matrix of beams column with rigid zone computer program for Newmark method computer program for Wilson method computer program for CQC method Goel steel-bracing hysteresis model and computer program Takeda model for RC columns and beams and computer program Cheng-Mertz model for bending coupling with shear and low-rise shear walls and computer program Cheng-Lou axial hysteresis model for RC columns and walls and computer program.

Solitons and almost-intertwining matrices

Abstract: We define the algebraic variety of almost intertwining matrices to be the set of triples (X,Y,Z) of n×n matrices for which XZ=YX+T for a rank one matrix T. A surprisingly simple formula is given for tau functions of the KP hierarchy in terms of such triples. The tau functions produced in this way include the soliton and vanishing rational solutions. The induced dynamics of the eigenvalues of the matrix X are considered, leading in special cases to the Ruijsenaars–Schneider particle system.

Jordan canonical form of a partitioned complex matrix and its application to real quaternion matrices

Abstract: Let Σ be the collection of all 2n × 2n partitioned complex matrices where A 1 and A 2 are n × n complex matrices, the bars on top of them mean matrix conjugate. We show that Σ is closed under similarity transformation to Jordan (canonical) forms. Precisely, any matrix in Σ is similar to a matrix in the form J ⊕ ∈ Σ via an invertible matrix in Σ, where J is a Jordan form whose diagonalelements all have nonnegative imaginary parts. An application of this result gives the Jordan form of real quaternion matrices.

Basic linear algebra

Abstract: Preface.- Foreward.- The Algebra of Matrices.- Some Applications of Matrices.- Systems of Linear Equations.- Invertible Matrices.- Vector Spaces.- Linear Mappings.- The Matrix Connection.- Determinants.- Eigenvalues and Eigenvectors.- The Minimal Polynomial.- Solutions to the Exercises.- Index.

Self-dual codes over F/sub p/ and weighing matrices

Abstract: Previously, self-dual codes have been constructed from Hadamard matrices. In this correspondence, codes constructed from weighing matrices, and in particular conference matrices are presented. A necessary condition for these codes to be self-dual is given, and examples are given for lengths up to 40. Codes constructed from all weighing matrices of order n/spl les/13 are also considered.

Non-Hermitian Random Matrices and the Calogero-Sutherland Model

Abstract: We study the statistical properties of the eigenvalues of non-Hermitian operators associated with the dissipative complex systems. By considering the Gaussian ensembles of such operators, a hierarchical relation between the correlators is obtained. Further, the eigenvalues are found to behave like particles moving on a complex plane under two-body (inverse square) and three-body interactions and there seems to underlie a deep connection and universality in the spectral behavior of different complex systems.

Normal matrices and polar decompositions indefinite inner products

Abstract: Normal matrices with respect to indefinite inner products are studied using the additive decomposition into selfadjoint and skewadjoint parts. In particular, several structural properties of indecomposable normal matrices are obtained. These properties are used to describe classes of matrices that are logarithms of selfadjoint or normal matrices. In turn, we use logarithms of normal matrices to study polar decompositions with respect to indefinite inner products. It is proved, in particular, that every normal matrix with respect to an indefinite inner product defined by an invertible Hermitian matrix having at most two negative (or at most two positive) eigenvalues, admits a polar decomposition. Previously known descriptions of indecomposable normals in indefinite inner products with at most two negative eigenvalues play a key role in the proof. Both real and complex cases are considered.

Preconditioning sparse matrices for computing eigenvalues and solving linear systems of equations

Abstract: Preconditioning Sparse Matrices for Computing Eigenvalues and Solving Linear Systems of Equations

A method for linear elasto-static analysis of multi-layered axisymmetrical bodies using Hankel's transform

Abstract: A method to determine the distribution of stresses and displacements in an infinite, linear, elastic, multi-layered medium subjected to static axisymmetric loading is presented in this work. By using axisymmetric governing equations, Hankel's transform and matrix analysis, the methodology gives a clearly arranged way to calculate the stresses and displacements in the medium. A numerical method for Hankel's transform is employed to perform the calculation. Two representative examples are studied. The results can be utilized as a fundamental solution for boundary element methods for the linear, elasto-static, axisymmetric multi-layered problem with a little modification.

Design of Generic Direct Sparse Linear System Solver in C++ for Power System Analysis

Abstract: This article presents the design of the generic linear system solver (LSS) for a class of large sparse symmetric matrices over real and complex numbers. These matrices correspond to one of the following: (1) symmetric positive definite (SPD) matrices, (2) complex Hermitian matrices, or (3) complex matrices with SPD real and imaginary matrices. Such matrices arise in various power system analysis applications like load flow analysis and short circuit analysis. A template facility of C++ is used to write a generic program on float, double, and complex data types. The design of the algorithm guarantees numerical stability and efficient sparsity implementation. A reusable class SET is defined to cater to graph theoretic computations. LSS problems with matrices up to 20,000 nodes have been tested. Another feature of the proposed LSS is the implementation of associative array, which allows subscripting an array with character strings, such as bus names. This helps in making the four power system analysis software user friendly. The proposed LSS reflects an important development towards a truly object-oriented four power system analysis software.

Construction of Matrices with Prescribed Singular Values and Eigenvalues

Abstract: Two issues concerning the construction of square matrices with prescribe singular values an eigenvalues are addressed. First, a necessary and sufficient condition for the existence of an n × n complex matrix with n given nonnegative numbers as singular values an m (≤ n) given complex numbers to be m of the eigenvalues is determined. This extends the classical result of Weyl and Horn treating the case when m = n. Second, an algorithm is given to generate a triangular matrix with prescribe singular values an eigenvalues. Unlike earlier algorithms, the eigenvalues can be arranged in any prescribe order on the diagonal. A slight modification of this algorithm allows one to construct a real matrix with specified real an complex conjugate eigenvalues an specified singular values. The construction is done by multiplication by diagonal unitary matrices, permutation matrices and rotation matrices. It is numerically stable and may be useful in developing test software for numerical linear algebra packages.

Deflation for block eigenvalues of block partitioned matrices with an application to matrix polynomials of commuting matrices

Abstract: A method for computing a complete set of block eigenvalues for a block partitioned matrix using a generalized form of Wielandt's deflation is presented. An application of this process is given to compute a complete set of solvents of matrix polynomials where the coefficients and the variable are commuting matrices.

Data Perturbations of Matrices of Pairwise Comparisons

Abstract: This paper deals with data perturbations of pairwise comparison matrices (PCM). Transitive and symmetrically reciprocal (SR) matrices are defined. Characteristic polynomials and spectral properties of certain SR perturbations of transitive matrices are presented. The principal eigenvector components of some of these PCMs are given in explicit form. Results are applied to PCMs occurring in various fields of interest, such as in the analytic hierarchy process (AHP) to the paired comparison matrix entries of which are positive numbers, in the dynamic input–output analysis to the matrix of economic growth elements of which might become both positive and negative and in vehicle system dynamics to the input spectral density matrix whose entries are complex numbers.

Signatures of quantum chaos in rare-earth elements: I. Characterization of the Hamiltonian matrices and coupling matrices of Ce I and Pr I using the statistical predictions of Random Matrix Theory

Abstract: Using the relativistic configuration interaction Hartree-Fock method the Hamiltonian matrices of Ce I, J = 4±, and Pr I, J = 11/2±, are studied. These matrices can be characterized as sparse, banded matrices, with a leading diagonal. Diagonalization of the Hamiltonian results in a set of energy eigenvalues and corresponding eigenvectors and the purpose of this investigation will be to characterize the Hamiltonian matrices and coupling matrices of Ce I and Pr I, for both ls and jj coupling representations, using various statistical predictions of Random Matrix Theory.

Eigenvalue estimates for Hankel matrices

Abstract: Positive-definite Hankel matrices have an important property: the ratio of the largest and the smallest eigenvalues (the spectral condition number) has as a lower bound an increasing exponential of the order of the matrix that is independent of the particular matrix entries. The proof of this fact is related to the so-called Vandermonde factorizations of positive-definite Hankel matrices. In this paper the structure of these factorizations is studied for real sign-indefinite strongly regular Hankel matrices. Some generalizations of the estimates of the spectral condition number are suggested.

Matrix Algebra for Applied Economics

Abstract: List of Chapters. Preface. BASICS. Introduction. Basic Matrix Operations. Special Matrices. Determinants. Inverse Matrices. NECESSARY THEORY. Linearly (IN)Dependent Vectors. Rank. Canonical Forms. Generalized Inverses. Solving Linear Equations. Eigenroots and Eigenvectors. Miscellanea. WORKING WITH MATRICES. Applying Linear Equations. Regression Analysis. Linear Statistical Models. Linear Programming. Markov Chain Models. References. Index.

Products of Real Matrices with Prescribed Characteristic Polynomials

Abstract: Let A be a matrix with entries in the field of real numbers. In this paper we give necessary and sufficient conditions for the existence of real matrices B and C, with prescribed characteristic polynomials, such that A=BC.