An Imbedding Method for Computing the Generalized Inverses

This paper deals with a system of ordinary differential equations with known conditions associated with a given matrix. By using analytical and computational methods, the generalized inverses of the given matrix can be determined. Among these are the weighted Moore-Penrose inverse, the Moore-Penrose inverse, the Drazin inverse and the group inverse. In particular, a new insight is provided into the finite algorithms for computing the generalized inverse and the inverse.     

Analytical aspects of computational linear algebra

This paper deals with a system of ordinary differential equations with known initial conditions associated with a given square matrix. By using standard analytical and computational methods many of the important aspects of the given matrix can be determined. Among these are its determinant, its adjoint, its inverse (if it exist), the coefficients of the characteristic polynomial, the location of the roots ofthe characteristic polynomial, and the corresponding eigenvectors. Concomitently, the differential system yields a treatment of inhomogeneous linear algebraic systems associated with the given matrix, as in economic input-output analysis. In particular, new insights are provided into Faddeev's algorithm for the coefficients of the characteristic polynomial.     

Maximum principle for optimal control of distributed-parameter systems with singular mass matrix

A maximum principle for the open-loop optimal control of a vibrating system relative to a given convex index of performance is investigated. Though maximum principles have been studied by many people (see, e.g., Refs. 1–5), the principle derived in this paper is of particular use for control problems involving mechanical structures. The state variable satisfies general initial conditions as well as a self-adjoint system of partial differential equations together with a homogeneous system of boundary conditions. The mass matrix is diagonal, constant, and singular, and the viscous damping matrix is diagonal. The maximum principle relates the optimal control with the solution of the homogeneous adjoint equation in which terminal conditions are prescribed in terms of the terminal values of the optimal state variable. An application of this theory to a structural vibrating system is given in a companion paper (Ref. 6).     

A natural order in dynamical systems based on Conley–Markov matrices

We introduce a natural order to study properties of dynamical systems, especially their invariant sets. The new concept is based on the classical Conley index theory and transition probabilities among neighborhoods of different invariant sets when the dynamical systems are perturbed by white noises. The transition probabilities can be determined by the Fokker–Planck equation and they form a matrix called a Markov matrix. In the limiting case when the random perturbation is reduced to zero, the Markov matrix recovers the information given by the Conley connection matrix. The Markov matrix also produces a natural order from the least to the most stable invariant sets for general dynamical systems. In particular, it gives the order among the local extreme points if the dynamical system is a gradient-like flow of an energy functional. Consequently, the natural order can be used to determine the global minima for gradient-like systems. Some numerical examples are given to illustrate the Markov matrix and its properties.     

The G-biliaison class of symmetric determinantal schemes

We consider a family of schemes, that are defined by minors of a homogeneous symmetric matrix with polynomial entries. We assume that they have maximal possible codimension, given the size of the matrix and of the minors that define them. We show that these schemes are G-bilinked to a linear variety of the same dimension. In particular, they can be obtained from a linear variety by a finite sequence of ascending G-biliaisons on some determinantal schemes. We describe the biliaisons explicitly in the proof of Theorem 2.3. In particular, it follows that these schemes are glicci.     

The number of harmonic frames of prime order

Harmonic frames of prime order are investigated. The primary focus is the enumeration of inequivalent harmonic frames, with the exact number given by a recursive formula. The key to this result is a one-to-one correspondence developed between inequivalent harmonic frames and the orbits of a particular set. Secondarily, the symmetry group of prime order harmonic frames is shown to contain a subgroup consisting of a diagonal matrix as well as a permutation matrix, each of which is dependent on the particular harmonic frame in question.     

Optimal actuators placements for the active control of flexible structures

A scheme for the optimal spatial placement of a limited number of sensors and actuators under a minimum energy requirement for the active control of flexible structures is proposed. The method is based on the interpretation of the functional relationship (transfer matrix/conrol influence matrix) between the actuators and modes of the structural system. It is shown that, from the form of the matrix, the controllability and observability of the system with respect to differing locations of the sensors and actuators can be established. The algorithm presented circumvents prevailing problems encountered in contemporary optimal control applications. In particular, and in order to enhance the results presented in this paper, numerical simulation for a prismatic beam subjected to horizontal random wind loads and a simply supported square plate modelled as a single degree of freedom system are given to illustrate the placement strategy.     

Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions

By taking as a “prototype problem” a one-delay linear autonomous system of delay differential equations we present the problem of computing the characteristic roots of a retarded functional differential equation as an eigenvalue problem for a derivative operator with non-local boundary conditions given by the particular system considered. This theory can be enlarged to more general classes of functional equations such as neutral delay equations, age-structured population models and mixed-type functional differential equations.It is thus relevant to have a numerical technique to approximate the eigenvalues of derivative operators under non-local boundary conditions. In this paper we propose to discretize such operators by pseudospectral techniques and turn the original eigenvalue problem into a matrix eigenvalue problem. This approach is shown to be particularly efficient due to the well-known “spectral accuracy” convergence of pseudospectral methods. Numerical examples are given.     

Oblique projectors and group involutory matrices

In this work, conditions characterizing when a lineal combination of two projectors of the same order is a group involutory matrix are analyzed. In particular, a representation of an involutory matrix as a sum or difference of two projectors is given. In addition, some results are applied to control problems. An application to obtain a state feedback control is given.     

Solution of a multiple Nevanlinna–Pick problem for Carathéodory functions via orthogonal rational functions in the matrix case

We consider an interpolation problem of Nevanlinna–Pick type for matrix‐valued Carathéodory functions, where the values of the functions and its derivatives up to certain orders are given at finitely many points of the open unit disk. For the non‐degenerate case, i.e., in the particular situation that a specific block matrix (which is formed by the given data in the problem) is positive Hermitian, the solution set of this problem is described in terms of orthogonal rational matrix‐valued functions. These rational matrix functions play here a similar role as Szegő's orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we present and use a connection between Szegő and Schur parameters for orthogonal rational matrix‐valued functions which in the primary situation of orthogonal polynomials was found by Geronimus. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Expected Conditioning

A perturbation analysis based on probabilistic arguments isdeveloped for a range of problems in numerical linear algebra,including well-determined and over-determined linear systems.Condition matrices and condition numbers are determined forthe expected value of the actual condition number of a problem.These enable attainable lower and upper bounds on the expectedcondition properties of a matrix to be given, independent ofany particular linear system. These estimates are much morereliable than those derived from conventional norm conditionnumbers, and are shown to reveal features which the latter cannot. The expected condition analysis has desirable properties underscaling transformations which is not the case for the norm conditionanalysis. It is shown that an optimal (or natural) scaling canbe associated with any matrix which moreover is readily computed.This enables the equilibration of a matrix to be carried out.Because this process is a uniquely defined projection it alwaysenables the best conditioned of all the possible equilibratedmatrices to be determined.     

Generalized Gram Matrix and Its Application to the Observability Problem for Linear Nonsteady Systems

Sufficient conditions for the nondegeneracy of a generalized Gram matrix are obtained. In particular, it is shown that the generalized Gram matrix is nondegenerate for the Chebyshev systems of functions. An application of the results to the observability problems for linear nonsteady systems of ordinary differential equations are given. In terms of the observability matrix, necessary and sufficient conditions of the complete and total observability by means of finite-parameter solving operations are established.     

Groups and semigroups generated by a single unitary orbit

Semigroup Forum - We investigate the structure of the multiplicative semigroup generated by the set of matrices that are unitarily equivalent to a given invertible matrix A. In particular, we give...     

On some methods for entropy maximization and matrix scaling

We describe and survey in this paper iterative algorithms for solving the discrete maximum entropy problem with linear equality constraints. This problem has applications e.g. in image reconstruction from projections, transportation planning, and matrix scaling. In particular we study local convergence and asymptotic rate of convergence as a function of the iteration parameter. For the trip distribution problem in transportation planning and the equivalent problem of scaling a positive matrix to achieve a priori given row and column sums, it is shown how the iteration parameters can be chosen in an optimal way. We also consider the related problem of finding a matrix X, diagonally similar to a given matrix, such that corresponding row and column norms in X are all equal. Reports of some numerical tests are given.     

Separation theorems for singular values of matrices and their applications in multivariate analysis

Separation theorems for singular values of a matrix, similar to the Poincaré separation theorem for the eigenvalues of a Hermitian matrix, are proved. The results are applied to problems in approximating a given r.v. by an r.v. in a specified class. In particular, problems of canonical correlations, reduced rank regression, fitting an orthogonal random variable (r.v.) to a given r.v., and estimation of residuals in the Gauss-Markoff model are discussed. In each case, a solution is obtained by minimizing a suitable norm. In some cases a common solution is shown to minimize a wide class of norms known as unitarily invariant norms introduced by von Neumann.     

Tikhonov solutions of approximately given systems of linear algebraic equations under finite perturbations of their matrices

The properties of a mathematical programming problem that arises in finding a stable (in the sense of Tikhonov) solution to a system of linear algebraic equations with an approximately given augmented coefficient matrix are examined. Conditions are obtained that determine whether this problem can be reduced to the minimization of a smoothing functional or to the minimal matrix correction of the underlying system of linear algebraic equations. A method for constructing (exact or approximately given) model systems of linear algebraic equations with known Tikhonov solutions is described. Sharp lower bounds are derived for the maximal error in the solution of an approximately given system of linear algebraic equations under finite perturbations of its coefficient matrix. Numerical examples are given.     

The Bezoutian and the eigenvalue-separation problem for matrix polynomials

A generalized Bezout matrix for a pair of matrix polynomials is studied and, in particular, the structure of its kernel is described and the relations to the greatest common divisor of the given matrix polynomials are presented. The classical root-separation problems of Hermite, Routh-Hurwitz and Schur-Cohn are solved for matrix polynomials in terms of this Bezout matrix. The eigenvalue-separation results are also expressed in terms of Hankel matrices whose entries are Markov parameters of rational matrix function. Some applications of Jacobi's method to these problems are pointed out.     

最小多项式矩阵与线性多变量系统(Ⅰ)

本文分为两部分:(Ⅰ)为关于最小多项式矩阵的理论:(Ⅱ)为最小多项式矩阵理论在线性多变量系统中的应用.在(Ⅰ)中,我们给出了线性变换在向量组的消失多项式矩阵与最小多项式矩阵的概念,给出了不变子空间的生成组与最小生成组的概念.在讨论了这些概念的基本性质之后,我们研究了它们与线性变换在任何不变子空间上诱导算子对应的特征矩阵之间的关系,给出了向量组的最小多项式矩阵类的特征,并给出了有相同生成空间的生成组之间的充分必要条件.利用这些结果,对于给定的矩阵A,给出了能使系统x=Ax+Bu完全可控的矩阵B的全体的集合的表达式.     

Multivariate Frobenius–Padé approximants: Properties and algorithms

The aim of this paper is to construct rational approximants for multivariate functions given by their expansion in an orthogonal polynomial system. This will be done by generalizing the concept of multivariate Padé approximation. After defining the multivariate Frobenius–Padé approximants, we will be interested in the two following problems: the first one is to develop recursive algorithms for the computation of the value of a sequence of approximants at a given point. The second one is to compute the coefficients of the numerator and denominator of the approximants by solving a linear system. For some particular cases we will obtain a displacement rank structure for the matrix of the system we have to solve. The case of a Tchebyshev expansion is considered in more detail.