Decomposable critical tensors

Let λ = (λ₁, … , λ_t) be a partition of m and its conjugate partition. Denote also by λ the irreducible C-character of S_m associated with λ. Let V be a finite dimensional vector space over C.The reach of an element of the symmetry class of tensors V_λ (symmetry class of tensors associated with λ) is defined. The concept of critical element is introduced, as an element whose reach has dimension equal to . It is observed that, in ∧^mV, the notions of critical element and decomposable element coincide. Known results for decomposable elements of ∧^mV are extended to critical elements of V_λ. In particular, for a basis of ⊗^mV induced by a basis of V, generalized Plücker polynomials are constructed in a way that the set of their common roots contains the set of the families of components of decomposable critical elements of V_λ.     

Additive preservers of non-zero decomposable tensors

Let T be an additive mapping from a tensor product of vector spaces over a field into itself. We describe T for the following two cases: (i) T is surjective and sends non-zero decomposable elements to non-zero decomposable elements, and (ii) T(A) is a non-zero decomposable element if and only if A is a non-zero decomposable element.     

Real linear Kronecker product operations

The Kronecker product in the real linear matrix analytic setting is studied. More versatile operations are proposed. Such generalizations are of interest for the same reasons the standard Kronecker product is. To give an example, new preconditioning ideas are suggested. In connection with this, several formulae for the inverse are devised. Orthogonal decompositions of real-entried matrices are derived through introducing new Kronecker product SVDs. Matrix equations are given to illustrate how the Kronecker product structures introduced can arise.     

A general product of tensors with applications

We study a general product of two n  -dimensional tensors A$\mathbb{A}$ and B$\mathbb{B}$ with orders m?2$m?2$ and k?1$k?1$. This product satisfies the associative law, and is a generalization of the usual matrix product. Using this product, many concepts and known results of tensors can be simply expressed and/or proved, and a number of applications of it will be given. Using the associative law of this tensor product and some properties on the resultant of a system of homogeneous equations on n variables, we define the similarity and congruence of tensors (which are also the generalizations of the corresponding relations for matrices), and prove that similar tensors have the same characteristic polynomials, thus the same spectra. We study two special kinds of similarity: permutational similarity and diagonal similarity, and their applications in the study of the spectra of hypergraphs and nonnegative irreducible tensors. We also define the direct product of tensors (in matrix case it is also called the Kronecker product), and give its applications in the study of the spectra of two kinds of the products of hypergraphs. We also give applications of this general product in the study of nonnegative tensors, including a characterization of primitive tensors, the upper bounds of primitive degrees and the cyclic indices of some nonnegative irreducible tensors.     

On some properties of the determinants of tensors

We use the Hilbert?s Nullstellensatz (Hilbert?s Zero Point Theorem) to give a direct proof of the formula for the determinants of the products of tensors. By using this determinant formula and using tensor product to represent the transformations of the slices of tensors, we prove some basic properties of the determinants of tensors which are the generalizations of the corresponding properties of the determinants for matrices. We also study the determinants of tensors after two types of transposes. We use the permutational similarity of tensors to discuss the relation between weakly reducible tensors and the triangular block tensors, and give a canonical form of the weakly reducible tensors.     

Deflating quadratic matrix polynomials with structure preserving transformations

Given a pair of distinct eigenvalues (λ₁,λ₂) of an n×n quadratic matrix polynomial Q(λ) with nonsingular leading coefficient and their corresponding eigenvectors, we show how to transform Q(λ) into a quadratic of the form having the same eigenvalue s as Q(λ), with Q_d(λ) an (n-1)×(n-1) quadratic matrix polynomial and q(λ) a scalar quadratic polynomial with roots λ₁ and λ₂. This block diagonalization cannot be achieved by a similarity transformation applied directly to Q(λ) unless the eigenvectors corresponding to λ₁ and λ₂ are parallel. We identify conditions under which we can construct a family of 2n×2n elementary similarity transformations that (a) are rank-two modifications of the identity matrix, (b) act on linearizations of Q(λ), (c) preserve the block structure of a large class of block symmetric linearizations of Q(λ), thereby defining new quadratic matrix polynomials Q₁(λ) that have the same eigenvalue s as Q(λ), (d) yield quadratics Q₁(λ) with the property that their eigenvectors associated with λ₁ and λ₂ are parallel and hence can subsequently be deflated by a similarity applied directly to Q₁(λ). This is the first attempt at building elementary transformations that preserve the block structure of widely used linearizations and which have a specific action.     

Computing multiple integrals involving matrix exponentials

In this paper, a generalization of a formula proposed by Van Loan [Computing integrals involving the matrix exponential, IEEE Trans. Automat. Control 23 (1978) 395–404] for the computation of multiple integrals of exponential matrices is introduced. In this way, the numerical evaluation of such integrals is reduced to the use of a conventional algorithm to compute matrix exponentials. The formula is applied for evaluating some kinds of integrals that frequently emerge in a number classical mathematical subjects in the framework of differential equations, numerical methods and control engineering applications.     

Palindromic matrix polynomials, matrix functions and integral representations

We study the properties of palindromic quadratic matrix polynomials φ(z)=P+Qz+Pz², i.e., quadratic polynomials where the coefficients P and Q are square matrices, and where the constant and the leading coefficients are equal. We show that, for suitable choices of the matrix coefficients P and Q, it is possible to characterize by means of φ(z) well known matrix functions, namely the matrix square root, the matrix polar factor, the matrix sign and the geometric mean of two matrices. Finally we provide some integral representations of these matrix functions.     

Detecting hyperbolic and definite matrix polynomials

Hyperbolic or more generally definite matrix polynomials are important classes of Hermitian matrix polynomials. They allow for a definite linearization and can therefore be solved by a standard algorithm for Hermitian matrices. They have only real eigenvalues which can be characterized as minmax and maxmin values of Rayleigh functionals, but there is no easy way to test if a given polynomial is hyperbolic or definite or not. Taking advantage of the safeguarded iteration which converges globally and monotonically to extreme eigenvalues we obtain an efficient algorithm that identifies hyperbolic or definite polynomials and enables the transformation to an equivalent definite linear pencil. Numerical examples demonstrate the efficiency of the approach.     

Existence of real eigenvalues of real tensors

We use the Brouwer degree to establish the existence of real eigenpairs of higher order real tensors in various settings. Also, we provide some finer criteria for the existence of real eigenpairs of two-dimensional real tensors and give a complete classification of the Brouwer degree zero and ±2 maps induced by general third order two-dimensional real tensors.     

On the quadratic two-parameter eigenvalue problem and its linearization

We introduce the quadratic two-parameter eigenvalue problem and linearize it as a singular two-parameter eigenvalue problem. This, together with an example from model updating, shows the need for numerical methods for singular two-parameter eigenvalue problems and for a better understanding of such problems.There are various numerical methods for two-parameter eigenvalue problems, but only few for nonsingular ones. We present a method that can be applied to singular two-parameter eigenvalue problems including the linearization of the quadratic two-parameter eigenvalue problem. It is based on the staircase algorithm for the extraction of the common regular part of two singular matrix pencils.     

Approximation by matrices with restricted spectra

We investigate the properties of the approximation of a matrix by matrices whose spectra are in a closed convex set of the complex plane. We explain why the Khalil and Maher characterization of an approximant, which spectrum is in a strip, is not quite correct. We prove that their characterization is valid but for another kind of approximation. We formulate a conjecture which leads to some algorithm for computing approximants. The conjecture is motivated by numerical experiments and some theoretical considerations. Separately we consider the approximation of normal matrices.     

How real is your matrix?

For scalars there is essentially just one way to define reality, real part and to measure nonreality. In this paper various ways of defining respective concepts for complex-entried matrices are considered. In connection with this, products of circulant and diagonal matrices often appear and algorithms to approximate additively and multiplicatively with them are devised. Multiplicative structures have applications, for instance, in diffractive optics, preconditioning and fast Fourier expansions.     

Twisted inner products and contraction inequalities on spaces of contravariant and covariant tensors

Given positive integers n and p, and a complex finite dimensional vector space V, we let S_n,p(V) denote the set of all functions from V×V×?×V-(n+p copies) to C that are linear and symmetric in the first n positions, and conjugate linear symmetric in the last p positions. Letting κ=min{n,p} we introduce twisted inner products, [·,·]_s,t,1?s,t?κ, on S_n,p(V), and prove the monotonicity condition [F,F]_s,t?[F,F]_u,v is satisfied when s?u?κ,t?v?κ, and F∈S_n,p(V). Using the monotonicity condition, and the Cauchy-Schwartz inequality, we obtain as corollaries many known inequalities involving norms of symmetric multilinear functions, which in turn imply known inequalities involving permanents of positive semidefinite Hermitian matrices. New tensor and permanental inequalities are also presented. Applications to partial differential equations are indicated.     

The general coupled matrix equations over generalized bisymmetric matrices

In the present paper, by extending the idea of conjugate gradient (CG) method, we construct an iterative method to solve the general coupled matrix equations

     

An algorithm for determining copositive matrices

In this paper, we present an algorithm of simple exponential growth called COPOMATRIX for determining the copositivity of a real symmetric matrix. The core of this algorithm is a decomposition theorem, which is used to deal with simplicial subdivision of on the standard simplex Δ_m, where each component of the vector β is −1, 0 or 1.     

On algebraic Riccati equations associated with M-matrices

We consider the algebraic Riccati equation for which the four coefficient matrices form an M-matrix K. When K is a nonsingular M-matrix or an irreducible singular M-matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M-matrix. Under a regularity assumption on the M-matrix K, we show that the Riccati equation still has a minimal nonnegative solution. We also study the properties of this particular solution and explain how the solution can be found by existing methods.     

Some properties of generalized K-centrosymmetric H-matrices

Every n×n$n\times n$ generalized K-centrosymmetric matrix A   can be reduced into a 2×2$2\times 2$ block diagonal matrix (see [Z. Liu, H. Cao, H. Chen, A note on computing matrix–vector products with generalized centrosymmetric (centrohermitian) matrices, Appl. Math. Comput. 169 (2) (2005) 1332–1345]). This block diagonal matrix is called the reduced form of the matrix A. In this paper we further investigate some properties of the reduced form of these matrices and discuss the square roots of these matrices. Finally exploiting these properties, the development of structure-preserving algorithms for certain computations for generalized K-centrosymmetric H-matrices is discussed.     

Regularized fast multiple-image deconvolution for LBT

In this paper we study a fast deconvolution technique for the image restoration problem of the Large Binocular Telescope (LBT) interferometer. Since LBT provides several blurred and noisy images of the same object, it requires the use of multiple-image deconvolution methods in order to produce a unique image with high resolution. Hence the restoration process is basically a linear ill-posed problem, with overdetermined system and data corrupted by several components of noise.     

Algorithms for the computation of functionals defined on the solution of a discrete ill-posed problem

We study a linear, discrete ill-posed problem, by which we mean a very ill-conditioned linear least squares problem. In particular we consider the case when one is primarily interested in computing a functional defined on the solution rather than the solution itself. In order to alleviate the ill-conditioning we require the norm of the solution to be smaller than a given constant. Thus we are lead to minimizing a linear functional subject to two quadratic constraints. We study existence and uniqueness for this problem and show that it is essentially equivalent to a least squares problem with a linear and a quadratic constraint, which is easier to handle computationally. Efficient algorithms are suggested for this problem.