Showing papers on "Matrix analysis published in 1997"

Nonnegative Matrices and Applications

Abstract: This book provides an integrated treatment of the theory of nonnegative matrices (matrices with only positive numbers or zero as entries) and some related classes of positive matrices, concentrating on connections with game theory, combinatorics, inequalities, optimisation and mathematical economics. The wide variety of applications, which include price fixing, scheduling and the fair division problem, have been carefully chosen both for their elegant mathematical content and for their accessibility to students with minimal preparation. Many results in matrix theory are also presented. The treatment is rigorous and almost all results are proved completely. These results and applications will be of great interest to researchers in linear programming, statistics and operations research. The minimal prerequisites also make the book accessible to first-year graduate students.

Nature of driving force for protein folding : a result from analyzing the statistical potential

Abstract: Proteins fold into specific three dimensional structures to perform their diverse biological functions. It is now well established that for small proteins the information contained in the amino acid sequence is sufficient to determine the folded structure, which is the structure with minimum free energy [1]. Thus the native structure is dictated by the physical interactions between amino acids in the sequence, and understanding the nature of such interactions is crucial for protein structure prediction. As a protein contains thousands of atoms and interacts with huge number of water molecules, it is not feasible to calculate the free energy function from first principles. An often adapted practical approach is to derive a coarse grained potential (often on the level of amino acids) using the known structures in the existing protein data banks. In such an approach, the energy of a particular substructure in proteins is derived from the number of its appearances in the structure data bank via a Boltzmann factor [2 ‐ 4]. A classic example of such a statistical potential is the Miyazawa-Jernigan (MJ) matrix, a 20 3 20 inter-residue contact-energy matrix derived by Miyazawa and Jernigan [2,5]. This matrix tabulates the interaction strength between any two types of amino acids in proteins, and has been widely applied in protein design and folding simulations [6,7]. In this Letter, we apply a general method of matrix analysis, namely, eigenvalue decomposition, to the MJ matrix. The analysis reveals an intrinsic regularity of the MJ matrix, which yields basic information about the nature of the driving force for protein folding. We show that despite the complicated interactions in proteins, the major driving force is hydrophobic interaction and a force of demixing, the latter obeying Hildebrand’s solubility theory of simple liquids [8]. The result allows us to attribute the interactions responsible for folding to quantifiable properties of individual amino acids. These properties suggest further experimental tests, and can be used for analyzing sequence-structure relation. Eigenvalue decomposition is a general approach to analyzing matrices. A given N 3 N real symmetric ma

Nature of Driving Force for Protein Folding-- A Result From Analyzing the Statistical Potential

Abstract: Proteins fold into specific three dimensional structures to perform their diverse biological functions. It is now well established that for small proteins the information contained in the amino acid sequence is sufficient to determine the folded structure, which is the structure with minimum free energy [1]. Thus the native structure is dictated by the physical interactions between amino acids in the sequence, and understanding the nature of such interactions is crucial for protein structure prediction. As a protein contains thousands of atoms and interacts with huge number of water molecules, it is not feasible to calculate the free energy function from first principles. An often adapted practical approach is to derive a coarse grained potential (often on the level of amino acids) using the known structures in the existing protein data banks. In such an approach, the energy of a particular substructure in proteins is derived from the number of its appearances in the structure data bank via a Boltzmann factor [2 ‐ 4]. A classic example of such a statistical potential is the Miyazawa-Jernigan (MJ) matrix, a 20 3 20 inter-residue contact-energy matrix derived by Miyazawa and Jernigan [2,5]. This matrix tabulates the interaction strength between any two types of amino acids in proteins, and has been widely applied in protein design and folding simulations [6,7]. In this Letter, we apply a general method of matrix analysis, namely, eigenvalue decomposition, to the MJ matrix. The analysis reveals an intrinsic regularity of the MJ matrix, which yields basic information about the nature of the driving force for protein folding. We show that despite the complicated interactions in proteins, the major driving force is hydrophobic interaction and a force of demixing, the latter obeying Hildebrand’s solubility theory of simple liquids [8]. The result allows us to attribute the interactions responsible for folding to quantifiable properties of individual amino acids. These properties suggest further experimental tests, and can be used for analyzing sequence-structure relation. Eigenvalue decomposition is a general approach to analyzing matrices. A given N 3 N real symmetric ma

Almost hermitian random matrices : crossover from wigner-dyson to ginibre eigenvalue statistics

Abstract: By using the method of orthogonal polynomials, we analyze the statistical properties of complex eigenvalues of random matrices describing a crossover from Hermitian matrices characterized by the Wigner-Dyson statistics of real eigenvalues to strongly non-Hermitian ones whose complex eigenvalues were studied by Ginibre. Two-point statistical measures [as, e.g., spectral form factor, number variance, and small distance behavior of the nearest neighbor distance distribution $p(s)$] are studied in more detail. In particular, we found that the latter function may exhibit unusual behavior $p(s)\ensuremath{\propto}{s}^{5/2}$ for some parameter values.

A precise time-step integration method by step-response and impulsive-response matrices for dynamic problems

Abstract: In this paper, a precise time-step integration method for dynamic problems is presented. The second-order differential equations for dynamic problems are manipulated directly. A general damping matrix is considered. The transient responses are expressed in terms of the steady-state responses, the given initial conditions and the step-response and impulsive-response matrices. The steady-state responses for various types of excitations are readily obtainable. The computation of the step-response and impulsive-response matrices and their time derivatives are studied in this paper. A direct computation of these matrices using the Taylor series solutions is not efficient when the time-step size Δt is not small. In this paper, the recurrence formulae relating the response matrices at t=Δt to those at t=Δt/2 are constructed. A recursive procedure is proposed to evaluate these matrices at t=Δt from the matrices at t=Δt/2m. The matrices at t=Δt/2m are obtained from the Taylor series solutions. To improve the computational efficiency, the relations between the response matrices and their time derivatives are investigated. In addition, these matrices are expressed in terms of two symmetric matrices that can also be evaluated recursively. Besides, from the physical point of view, these matrices should be banded for small Δt. Both the stability and accuracy characteristics of the present algorithm are studied. Three numerical examples are used to illustrate the highly precise and stable algorithm. © 1997 John Wiley & Sons, Ltd.

The structure of sparse resultant matrices

Abstract: Resultants characterize the existence of roots of systems of multivariatc nonlinear polynomial equations, while their matrices reduce the computation of all common zeros to a problemi nlirmaralgebra. Sparse elimination theory ha.s introduced the sparse resultant, which takes into account the sparse structurr of the polynomials. The construction of sparse resultant, or Newton, matrices is a critical step in the computation of the resultant and the solution of the system. We exploit the matrix structure and decrease the time complexity of constructing such matrices to roughly quadratic inthe matrix dimension, whereas the previous methods had cubic complexity. The space complexity is also decreased by one order of magnitude. These results imply similar improvements in the complexity of computing the resultant itself and of solving zero-dimensional systems. We apply some novel techniques for determining the rank of rectangularmatrices byanexact or numerical computation. Finally, we improve theexisting complexity forpolynomid multiplication under our model of sparseness, offering bounds linear in the number of variables and the number of nonzero terms.

Convex invertible cones of state space systems

Abstract: In the paper [CL1] the notion of a convex invertible cone,cic, of matrices was introduced and its geometry was studied In that paper close connections were drawn between thiscic structure and the algebraic Lyapunov equation In the present paper the same geometry is extended to triples of matrices andcics of minimal state space models are defined and explored This structure is then used to study balancing, Hankel singular values, and simultaneous model order reduction for a set of systems State spacecics are also examined in the context of the so-called matrix sign function algorithm commonly used to solve the algebraic Lyapunov and Riccati equations

Universal cubic eigenvalue repulsion for random normal matrices

Abstract: Random matrix models consisting of normal matrices, defined by the sole constraint $[N^{\dag},N]=0$, will be explored. It is shown that cubic eigenvalue repulsion in the complex plane is universal with respect to the probability distribution of matrices. The density of eigenvalues, all correlation functions, and level spacing statistics are calculated. Normal matrix models offer more probability distributions amenable to analytical analysis than complex matrix models where only a model wth a Gaussian distribution are solvable. The statistics of numerically generated eigenvalues from gaussian distributed normal matrices are compared to the analytical results obtained and agreement is seen.

Fast inversion algorithms for diagonal plus semiseparable matrices

Abstract: Here are considered matrices represented as a sum of diagonal and semiseparable ones. These matrices belong to the class of structured matrices which arises in numerous applications. FastO(N) algorithms for their inversion were developed before under additional restrictions which are a source of instability. Our aim is to eliminate these restrictions and to develop reliable and stable numerical algorithms. In this paper we obtain such algorithms with the only requirement that the considered matrix is invertible and its determinant is not close to zero. The case of semiseparable matrices of order one was considered in detail in an earlier paper of the authors.

The V-density of eigenvalues of non symmetric random matrices and rigorous proof of the strong Circular law

Abstract: We review some results obtained in series of papers on non Hermitian random matrices in some problems of spin glasses and neural nets. We present new theory of such matrices on the basis of the V-transform of normalized spectral function (n.s.f.) i^n(^,2/) of the eigenvalues of non symmetric matrix Ξ with n.s.f. /xn(x,r) of the eigenvalues of the Hermitian G-matrix (Ξ τ/) (Ξ τ/)*, τ = t + is : 7 q φ. Ο, and the modified V^-transform:

Mass matrices by minimization of modal errors

Abstract: A new approach to constructing mass matrices is presented, based on expressing it through use of a variable parameter. This allows the mass matrix to be adjusted in such a way that a simple eigenvalue problem get the best solution possible in terms of some error measure. This procedure is used to create both diagonal mass matrices and mixed mass matrices. © 1997 John Wiley & Sons, Ltd.

The minimal eigenvalues of a class of block-tridiagonal matrices

Abstract: In this correspondence, we study the minimal eigenvalues of a class of block-tridiagonal matrices from telecommunication system analysis. We present an eigenvalue analysis for two-user systems and efficient estimates for m-user systems.

Computing minimal partial realizations via a Lanczos-type algorithm for multiple starting vectors

Abstract: We describe a Lanczos-type procedure that reduces a given realization of a finite sequence of (moment) matrices to a minimal partial realization. A key feature of this procedure is that the underlying Lanczos-type algorithm is directly applied to the matrix triplet describing the given realization, rather than to the moment matrices. It thus avoids explicit formulation of and the usually unstable computation with the moment matrices.

A new eigenvalue placement method for linear dynamic systems

Abstract: Eigenvalue placement in a stable subregion of the s-plane is investigated for multivariable linear dynamic systems. A theorem that relates the bounds of a semi-annular region to the LQ performance index weighting matrices is used to develop a design procedure to relocate the eigenvalues. An example is included to demonstrate the successful implementation of the new method.

Matrix-theoretical analysis in the Laplace domain for the time lags and mean first passage times for reaction-diffusion transport

Abstract: Siegel’s matrix analysis of membrane transport in the Laplace domain [J. Phys. Chem. 95, 2556 (1991)], which is restricted to zero initial distribution, has been extended to including the case of nonzero initial distribution. This extension leads to a more general transport equation with Siegel’s results as a special case. The new transport equation allows us to formulate the mean-first-passage time t for various boundary conditions, if the initial distribution is stipulated to be of the Dirac delta-function type; and the steady-state permeability P and time lag tL, if zero initial distribution is employed. Based on this matrix analysis we also propose an algorithm for quick and effective numerical computations of P, tL, and t. Examples are given to demonstrate the application of this algorithm, and the numerical results are compared with the theoretical ones. The validity of the transport equation is also checked by a Green’s function.

New algorithms for the derivation of the transfer-function matrices of 2-D state-space discrete systems

Abstract: New algorithms for the derivation of the transfer-function matrices of two-dimensional (2-D) discrete systems from the Roesser and Fornasini-Marchesini state-space models are presented. Two key steps in developing the algorithms are as follows. First, the transfer-function matrix is reformulated in terms of the characteristic polynomials of the matrices involved. Second, an efficient algorithm for the determination of 1-D polynomial coefficients is developed and is, in turn, used to determine the coefficient matrices of the 2-D transfer-function matrix. The proposed algorithms are computationally efficient and reliable. The efficiency of the algorithms is illustrated by comparing the proposed method with two existing methods through examples.