A matrix pencil approach to the existence of compactly supported reconstruction functions in average sampling

The aim of this work is to solve a question raised for average sampling in shift-invariant spaces by using the well-known matrix pencil theory. In many common situations in sampling theory, the available data are samples of some convolution operator acting on the function itself: this leads to the problem of average sampling, also known as generalized sampling. In this paper we deal with the existence of a sampling formula involving these samples and having reconstruction functions with compact support. Thus, low computational complexity is involved and truncation errors are avoided. In practice, it is accomplished by means of a FIR filter bank. An answer is given in the light of the generalized sampling theory by using the oversampling technique: more samples than strictly necessary are used. The original problem reduces to finding a polynomial left inverse of a polynomial matrix intimately related to the sampling problem which, for a suitable choice of the sampling period, becomes a matrix pencil. This matrix pencil approach allows us to obtain a practical method for computing the compactly supported reconstruction functions for the important case where the oversampling rate is minimum. Moreover, the optimality of the obtained solution is established.     

Factorization of matrix functions and their inverses via power product expansions

The conversion of a power series with matrix coefficients into an infinite product of certain elementary matrix factors is studied. The expansion of a power series with matrix coefficients as the inverse of an infinite product of elementary factors is also analyzed. Each elementary factor is the sum of the identity matrix and a certain matrix coefficient multiplied by a certain power of the variable. The two expansions provide us with representations of a matrix function and its inverse by infinite products of elementary factors. Estimates on the domain of convergence of the infinite products are given.     

Circular trichotomy of the spectrum of regular matrix pencils

We introduce a method for approximating the right and left deflating subspaces of a regular matrix pencil corresponding to the eigenvalues inside, on and outside the unit circle. The method extends the iteration used in the context of spectral dichotomy, where the assumption on the absence of eigenvalues on the unit circle is removed. It constructs two matrix sequences whose null spaces and the null space of their sum lead to approximations of the deflating subspaces corresponding to the eigenvalues of modulus less than or equal to 1, equal to 1 and larger than or equal to 1. An orthogonalization process is then used to extract the desired delating subspaces. The resulting algorithm is an inverse free, easy to implement, and sufficiently fast. The derived convergence estimates reveal the key parameters, which determine the rate of convergence. The method is tested on several numerical examples.     

A nonlinear variation of constants method for integro differential and integral equations

A variation of constants technique is utilized to obtain representation formulas for solutions of perturbed nonlinear integrodifferential and integral equations. These representations are used to analyze boundedness and stability properties of perturbed integral equations. Questions on the existence of the inverse of the fundamental matrix as well on the existence of the semigroup property of the fundamental matrix are discussed.     

An iterative method for solving the stable subspace of a matrix pencil and its application

This work is to propose an iterative method of choice to compute a stable subspace of a regular matrix pencil. This approach is to define a sequence of matrix pencils via particular left null spaces. We show that this iteration preserves a semigroup property depending only on the initial matrix pencil. Via this recursion relationship, we propose an accelerated iterative method to compute the stable subspace and use it to provide a theoretical result to solve the principal square root of a given matrix, both nonsingular and singular. We show that this method can not only find out the matrix square root, but also construct an iterative approach which converges to the square root with any desired order.     

A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix

Based on a quadratical convergence method, a family of iterative methods to compute the approximate inverse of square matrix are presented. The theoretical proofs and numerical experiments show that these iterative methods are very effective. And, more importantly, these methods can be used to compute the inner inverse and their convergence proofs are given by fundamental matrix tools.     

Nonnegative definite matrices and their applications to matrix quadratic programming problems

In this paper, the concept that a matrix is nonnegative definite over a subspace and the tool of generalized inverse are used to express a general form of matrix quadratic programming. Several fundamental conclusions are obtained. An application to the common penalty method for handling constrained minimization problem is given.     

Perturbation and robust stability of autonomous singular linear matrix difference equations

In the perturbation theory of linear matrix difference equations, it is well known that the theory of finite and infinite elementary divisors of regular matrix pencils is complicated by the fact that arbitrarily small perturbations of the pencil can cause them to disappear. In this paper, the perturbation theory of complex Weierstrass canonical form for regular matrix pencils is investigated. By using matrix pencil theory and the Weierstrass canonical form of the pencil we obtain bounds for the finite elementary divisors of a perturbed pencil. Moreover we study robust stability of a class of linear matrix difference equations (of first and higher order) whose coefficients are square constant matrices.     

Nonnegative definite matrices and their applications to matrix quadratic programming problems

In this paper, the concept that a matrix is nonnegative definite over a subspace and the tool of generalized inverse are used to express a general form of matrix quadratic programming. Several fundamental conclusions are obtained. An application to the common penalty method for handling constrained minimization problem is given.     

The canonical structure of a pencil of degenerate matrix functions

In this paper we study global properties of a pencil of identically degenerate matrix functions with a compact domain of definition. Matrix functions are assumed to have a constant rank and all roots of the characteristic equation of the matrix pencil are assumed to have a constant multiplicity at each point in the domain of definition. We obtain sufficient conditions for the smooth orthogonal similarity of matrix functions to the upper triangular form and sufficient conditions for the smooth equivalence of the pencil of matrix functions to its canonical form. We illustrate the obtained results with simple examples.     

A deflation method for regular matrix pencils

A generalization of the concept of eigenvalue is introduced for a matrix pencil and it is called eigenpencil; an eigenpencil is a pencil itself and it contains part of the spectral information of the matrix pencil. A Wielandt type deflation procedure for regular matrix pencils is developed, using eigenpencils and supposing that they can have both finite and infinite eigenvalues. A numerical example illustrates the proposed method.     

Identities via Bell matrix and Fibonacci matrix

In this paper, we study the relations between the Bell matrix and the Fibonacci matrix, which provide a unified approach to some lower triangular matrices, such as the Stirling matrices of both kinds, the Lah matrix, and the generalized Pascal matrix. To make the results more general, the discussion is also extended to the generalized Fibonacci numbers and the corresponding matrix. Moreover, based on the matrix representations, various identities are derived.     

Structural and computational properties of possibly singular semiseparable matrices

A classical result of structured numerical linear algebra states that the inverse of a nonsingular semiseparable matrix is a tridiagonal matrix. Such a property of a semiseparable matrix has been proved to be useful for devising linear complexity solvers, for establishing recurrence relations among its columns or rows and, moreover, for efficiently evaluating its characteristic polynomial. In this paper, we provide sparse structured representations of a semiseparable matrix A which hold independently of the fact that A is singular or not. These relations are found by pointing out the band structure of the inverse of the sum of A plus a certain sparse perturbation of minimal rank. Further, they can be used to determine in a computationally efficient way both a reflexive generalized inverse of A and its characteristic polynomial.     

Structured eigenvalue condition numbers and linearizations for matrix polynomials

This work is concerned with eigenvalue problems for structured matrix polynomials, including complex symmetric, Hermitian, even, odd, palindromic, and anti-palindromic matrix polynomials. Most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size. Recently, linearizations have been classified for which the pencil reflects the structure of the original polynomial. A question of practical importance is whether this process of linearization significantly increases the eigenvalue sensitivity with respect to structured perturbations. For all structures under consideration, we show that this cannot happen if the matrix polynomial is well scaled: there is always a structured linearization for which the structured eigenvalue condition number does not differ much. This implies, for example, that a structure-preserving algorithm applied to the linearization fully benefits from a potentially low structured eigenvalue condition number of the original matrix polynomial.     

Representations and properties of the W-Weighted Drazin inverse

Several new representations of the W-weighted Drazin inverse are introduced. These representations are expressed in terms of various matrix powers as well as in terms of matrix products involving the Moore–Penrose inverse and the usual matrix inverse. Also, the properties of various generalized inverses which arise from derived representations are investigated. The computational complexity and efficiency of the proposed representations are considered. Representations are tested and compared among themselves in a substantial number of randomly generated test examples.     

Estimation of the precision matrix of a singular Wishart distribution and its application in high-dimensional data

In this article, the Stein-Haff identity is established for a singular Wishart distribution with a positive definite mean matrix but with the dimension larger than the degrees of freedom. This identity is then used to obtain estimators of the precision matrix improving on the estimator based on the Moore-Penrose inverse of the Wishart matrix under the Efron-Morris loss function and its variants. Ridge-type empirical Bayes estimators of the precision matrix are also given and their dominance properties over the usual one are shown using this identity. Finally, these precision estimators are used in a quadratic discriminant rule, and it is shown through simulation that discriminant methods based on the ridge-type empirical Bayes estimators provide higher correct classification rates.     

A MODIFIED INVERSE ITERATION FOR A LARGE SPARSE SPD GENERALIZED EIGENPROBLEM

In this paper, an algorithm based on a shifted inverse power iteration for computing generalized eigenvalues with corresponding eigenvectors of a large scale sparse symmetric positive definite matrix pencil is presented. It converges globally with a cubic asymptotic convergence rate, preserves sparsity of the original matrices and is fully parallelizable. The algebraic multilevel itera-tion method (AMLI) is used to improve the efficiency when symmetric positive definite linear equa-tions need to be solved.     

A new updating method for the damped mass-spring systems

In this paper, we concern the inverse problem of constructing a monic quadratic pencil which possesses the prescribed partial eigendata, and the damping matrix and stiffness matrix are symmetric tridiagonal. Furthermore, the stiffness matrix is positive semi-definite and weakly diagonally dominant, which has positive diagonal elements and negative off-diagonal elements. Based on the solution of the inverse eigenvalue problem, we apply the alternating direction method with multiplier to solve the finite element model updating problem for the serially linked mass-spring system. The positive semi-definiteness of stiffness matrix, nonnegativity of stiffness and the physical connectivity of the original model are preserved. Numerical results show that our proposed method works well.     

On condition numbers of a nondefective multiple eigenvalue of a nonsymmetric matrix pencil

This paper considers the condition numbers of a nondefective multiple eigenvalue of a nonsymmetric matrix pencil. Based on the directional derivatives of a nondefective multiple eigenvalue of a nonsymmetric matrix pencil analytically dependent on several parameters, different condition numbers of a nondefective multiple eigenvalue are introduced. The computable expressions and bounds of introduced condition numbers are derived. Moreover, some results on the perturbation of a nondefective multiple eigenvalue of a nonsymmetric matrix pencil are given.