Polynomials in a matrix

Let A be a matrixp(x) a polynomial. Put B=p(A). It is shown that necessary and sufficient conditions for A to be a polynomial in B are (i) if λ is any eigenvalue of A, and if some elementary divisor of A corresponding to λ is nonlinear, thenp ^′(λ)≠0;and (ii) if λ,μ are distinct eigenvalues of A, then p(λ)p(μ) are also distinct. Here all computations are over some algebraically closed field.     

A gradient method for the matrix eigenvalue problemAx=λBx

LetA andB ben×m matrices. A gradient method for the minimization of the functionalF(x)=‖Ax?(〈Ax, Bx〉/〈Bx, Bx〉)Bx‖² is developed. The minima ofF are the eigenvectors of the eigenproblemAx=λBx. The concept of a non-defective eigenvalue for this generalized eigenvalue problem is developed. It is then shown that geometric convergence is attained for non-defective eigenvalues. A convergence rate analysis is given where it is shown that the rapidity of convergence of the gradient method to an eigenvalue λ depends on the degree of non-defectiveness of λ and the singular values ofA?λB.     

A note on the normwise perturbation theory for the regular generalized eigenproblem

In this paper, we present a normwise perturbation theory for the regular generalized eigenproblem Ax = λBx, when λ is a semi-simple and finite eigenvalue, which departs from the classical analysis with the chordal norm [9]. A backward error and a condition number are derived for a choice of flexible measure to represent independent perturbations in the matrices A and B. The concept of optimal backward error associated with an eigenvalue only is also discussed. © 1998 John Wiley & Sons, Ltd.     

On the spectra of a family of positive matrices

It is shown that every positive matrix A can be embedded in an analytic family of positive matrices {A(ν) : ν∈$\text{R}$} in such a way that A(1)=A, $\text{A(0)≡}\text{A}\text{?}$ is symmetric, and A(-1)=A^T. A necessary and sufficient condition that A and Å have the same maximal eigenvalue and that their ergodic limits have the same diagonal elements is stated and proved.     

Inverse kinematics solutions for industrial robot manipulators with offset wrists

In this paper, the inverse kinematics solutions for 16 industrial 6-Degrees-of-Freedom (DOF) robot manipulators with offset wrists are solved analytically and numerically based on the existence of the closed form equations. A new numerical algorithm is proposed for the inverse kinematics of the robot manipulators that cannot be solved in closed form. In order to illustrate the performance of the New Inverse Kinematics Algorithm (NIKA), the simulation results attained from NIKA are compared with those obtained from well-known Newton–Raphson Algorithm (NRA). The inverse kinematics solutions of two robot manipulators with offset wrists are given as examples. In order to have a complete idea, the inverse kinematics solution techniques for 16 industrial robot manipulators are also summarized in a table.     

On matrices leaving invariant a nontrivial convex set

A real square matrix A leaves a nontrivial convex set invariant if there exists a convex set C, which is not a linear subspace, such that A(C) ? C. It is shown that this is equivalent to the statement that A has an eigenvalue λ with λ?0 or |λ|?1.     

Spectral asymptotics of two parameter nonlinear Sturm-Liouville problem

In this paper we establish an asymptotic formula of n ? th variational eigenvalue λ = λ_n(μ,α) of the nonlinear Sturm-Liouville problem with two parameters on the general level set N _α,μ as μ → ∞.     

Piecewise linear maps, Liapunov exponents and entropy

Let be a class of piecewise linear maps associated with a transition matrix A. In this paper, we prove that if fA,x∈LA, then the Liapunov exponent λ(x) of fA,x is equal to a measure theoretic entropy hmA,x of fA,x, where mA,x is a Markov measure associated with A and x. The Liapunov exponent and the entropy are computable by solving an eigenvalue problem and can be explicitly calculated when the transition matrix A is symmetric. Moreover, we also show that maxxλ(x)=maxxhmA,x=log(λ₁), where λ₁ is the maximal eigenvalue of A.     

Particular formulae for the Moore-Penrose inverse of a columnwise partitioned matrix

An essential part of Cegielski’s [Obtuse cones and Gram matrices with non-negative inverse, Linear Algebra Appl. 335 (2001) 167-181] considerations of some properties of Gram matrices with nonnegative inverses, which are pointed out to be crucial in constructing obtuse cones, consists in developing some particular formulae for the Moore-Penrose inverse of a columnwise partitioned matrix A = (A₁ : A₂) under the assumption that it is of full column rank. In the present paper, these results are generalized and extended. The generalization consists in weakening the assumption mentioned above to the requirement that the ranges of A₁ and A₂ are disjoint, while the extension consists in introducing the conditions referring to the class of all generalized inverses of A.     

Characterizing shape theories by Kan extensions

Every pair (C,K) of categories, where K is a proreflective subcategory of C, generates a shape theory. As a main result in this paper we give a characterization of such pairs, showing that these are exactly those having the property that every functor F :K→A has a Kan extension Ran F :C→A, which is preserved by all functors commuting with inverse limits.     

On constructing the solutions to integral equations when ¦λ¦ > ¦λ1¦

A method is given for constructing the solution to Fredholm integral equations of the second kind for $\text{¦λ¦ < ¦λ}{}_{\mathrm{k}}\text{¦}$. Both the cases when λ is a regular value and when λ is an eigenvalue of the integral operator are considered.     

Convergence of inexact inverse iteration with application to preconditioned iterative solves

In this paper we study inexact inverse iteration for solving the generalised eigenvalue problem A x=λM x. We show that inexact inverse iteration is a modified Newton method and hence obtain convergence rates for various versions of inexact inverse iteration for the calculation of an algebraically simple eigenvalue. In particular, if the inexact solves are carried out with a tolerance chosen proportional to the eigenvalue residual then quadratic convergence is achieved. We also show how modifying the right hand side in inverse iteration still provides a convergent method, but the rate of convergence will be quadratic only under certain conditions on the right hand side. We discuss the implications of this for the preconditioned iterative solution of the linear systems. Finally we introduce a new ILU preconditioner which is a simple modification to the usual preconditioner, but which has advantages both for the standard form of inverse iteration and for the version with a modified right hand side. Numerical examples are given to illustrate the theoretical results. AMS subject classification (2000)  65F15, 65F10     

Polynomials satisfied by two linked matrices

Polynomials in two variables, evaluated at A and with A being a square complex matrix and being its transform belonging to the set {A⁼, A^†, A^∗}, in which A⁼, A^†, and A^∗ denote, respectively, any reflexive generalized inverse, the Moore-Penrose inverse, and the conjugate transpose of A, are considered. An essential role, in characterizing when such polynomials are satisfied by two matrices linked as above, is played by the condition that the column space of A is the column space of . The results given unify a number of prior, isolated results.     

S~5上仿Blaschke张量的特征值为常数的超曲面

设x:M→S~(n+1)是(n+1)-维单位球面上不含脐点的超曲面,在S~(n+1)的Moebius变换群下浸入x的四个基本不变量是:一个黎曼度量g称为Moebius度量;一个1-形式Φ称为Moebius形式;一个对称的(0,2)张量A称为Blaschke张量和一个对称的(0,2)张量B称为Moebius第二基本形式.对称的(0,2)张量D=A+λB也是Moebius不变量,其中λ是常数,D称为浸入x的仿Blaschke张量.李海中和王长平研究了满足条件:(i)Φ=0;(ii)A+λB+μg=0的超曲面,其中λ和μ都是函数,他们证明了λ和μ都是常数,并且给出了这类超曲面的分类,也就是在Φ=0的条件下D只有一个互异的特征值的超曲面的分类.本文对S~5上满足如下条件的超曲面进行了完全分类:(i)Φ=0,(ii)对某常数λ,D具有常数特征值.     

Lower bounds for the minimum eigenvalue of Hadamard product of an M-matrix and its inverse

For the Hadamard product A ° A⁻¹ of an M-matrix A and its inverse A⁻¹, we give new lower bounds for the minimum eigenvalue of A ° A⁻¹. These bounds are strong enough to prove the conjecture of Fiedler and Markham [An inequality for the Hadamard product of an M-matrix and inverse M-matrix, Linear Algebra Appl. 101 (1988) 1-8].     

用简单反迭代法计算三对角矩阵的特征向量

与特征值计算的算法丰富多彩相比,在已知比较精确的特征值的情况下,求其相应的特征向量的算法却不多见,已有的算法有基本反迭代法[1][2][4][5]、交替法[3]等.到目前为止,计算特征向量的算法都是基于反迭代法的,衡量算法是否收敛都是以残量的大小为标准,本文的算法也不例外.本文的目的就是计算不可约实对称三对角矩阵T=[bj-1,aj,bj]的相应于某个特征值λi（已得到其近似λ）的特征向量.首先我们来看下面的例子:例1 我们取T为201阶的Wilkinson负矩阵,λ取计算的最大特征值,分别令迭代的初始向量是e1,e100,e201,e=（1,1,…,1）T.图1反映了反迭代的收敛速度.     

Solution of the eigenvalue problem for a regular pencil λA0−A1 with singular matrices

One considers the generalized eigenvalue problem (A₀λ?A₁)x=0, (1) when one or both matrices A₀,A₁ are singular and ker A₀ ∩ ker A₁=φ is the empty set. With the aid of the normalized process, the solving of problem (1) reduces to the solving of the eigenvalue problem of a constant matrix of order r=min (r₀,r₁), where r₀,r₁ are the ranks of the matrices A₀,A₁, which are determined at the normalized decomposition of the matrices. One gives an Algol program which performs the presented algorithm and testing examples.     

The periodic and eigen solutions of the equation governing a nonrotating viscoelastic earth model under transient force

The problem of existence of the periodic solution of the equation governing a nonrotating viscoelastic earth model under transient force is examined. By first formulating the governing equations, using the methods of Coleman and Noll (Rev. Modern Physics33 (2) (1961), 239–249), Dahlen and Smith (Philos. Trans. Roy. Soc. London A279 (1975), 583–624), and Biot (“Mechanics of Incremental Deformations,” Wiley, New York, 1965), these equations are subjected to oscillatory displacement resulting in an eigenvalue problem whose solutions are the viscoelastic-gravitational displacement eigenfunctions U(x) with associated eigenfrequencies ω. A theorem is then proved to show the existence of a periodic solution.     

THE NET REPRODUCTIVE VALUE AND STABILITY IN MATRIX POPULATION MODELS

The net reproductive value n is defined for a general discrete linear population model with a non-negative projection matrix. This number is shown to have the biological interpretation of the expected number of offspring per individual over its life time. The main result relates n to the population's growth rate (i.e. the dominant eigenvalue λ of the projection matrix) and shows that the stability of the extinction state (the trivial equilibrium) can be determined by whether n is less than or greater than 1. Examples are given to show that explicit algebraic formulas for n are often derivable, and hence available for both numerical and parameter studies of stability, when no such formulas for λ are available.     

Properties of Hadamard product of inverse M‐matrices

This paper concerns with the properties of Hadamard product of inverse M‐matrices. Structures of tridiagonal inverse M‐matrices and Hessenberg inverse M‐matrices are analysed. It is proved that the product A ○ A^T satisfies Willoughby's necessary conditions for being an inverse M‐matrix when A is an irreducible inverse M‐matrix. It is also proved that when A is either a Hessenberg inverse M‐matrix or a tridiagonal inverse M‐matrix then A ○ A^T is an inverse M‐matrix. Based on these results, the conjecture that A ○ A^T is an inverse M‐matrix when A is an inverse M‐matrix is made. Unfortunately, the conjecture is not true. Copyright © 2004 John Wiley Sons, Ltd.