Generalized Matrix Functions Research Articles

Generalized matrix functions and Hermitian matrices

Generalized matrix functions and Hermitian matrices

Generalized matrix functions and some refinement inequalities

In this short paper, we provide some refinement inequalities on generalized matrix functions. In particular, permanent inequalities concerning doubly stochastic positive semidefinite matrices are also included.

Some remarks on permanental dominance conjecture

In this paper we provide an identity between the determinant and other generalized matrix functions, and give a criterion for positive semi-definite matrices to satisfy the permanental dominance conjecture. As a consequence, infinitely many classes of positive semi-definite matrices satisfying the conjecture (does not depend on groups or characters) are provided by generating from any positive semi-definite matrix having no zero in the first column.

Inverses of Toeplitz plus Hankel operators with generating matrix functions

Inverses of Toeplitz plus Hankel operators with generating matrix functions

Generalized matrix functions, permutation matrices and symmetric matrices

The purpose of this paper is to study generalized matrix functions only using the permutation matrices and symmetric matrices. Firstly the zeroness of a generalized matrix function and then the equality of two generalized matrix functions on the permutation matrices and symmetric matrices will be examined. Secondly generalized matrix functions preserving commutativity of the permutation matrices or commutativity of the symmetric matrices will be characterized. Thirdly generalized matrix functions which preserve product of the permutation matrices or product of the symmetric matrices will be investigated. Finally the Cayley-Hamilton Theorem for generalized characteristic polynomials using the permutation matrices and symmetric matrices will be studied.

Some one-variable identities on generalized matrix functions which imply determinant

Some one-variable identities on generalized matrix functions which imply determinant

Computation of generalized matrix functions with rational Krylov methods

We present a class of algorithms based on rational Krylov methods to compute the action of a generalized matrix function on a vector. These algorithms incorporate existing methods based on the Golub-Kahan bidiagonalization as a special case. By exploiting the quasiseparable structure of the projected matrices, we show that the basis vectors can be updated using a short recurrence, which can be seen as a generalization to the rational case of the Golub-Kahan bidiagonalization. We also prove error bounds that relate the error of these methods to uniform rational approximation. The effectiveness of the algorithms and the accuracy of the bounds is illustrated with numerical experiments.

Integral representations for higher-order Fréchet derivatives of matrix functions: Quadrature algorithms and new results on the level-2 condition number

We propose an integral representation for the higher-order Fréchet derivative of analytic matrix functions f(A) which unifies known results for the first-order Fréchet derivative of general analytic matrix functions and for higher-order Fréchet derivatives of A−1. We highlight two applications of this integral representation: On the one hand, it allows to find the exact value of the level-2 condition number (i.e., the condition number of the condition number) of f(A) for a large class of functions f when A is Hermitian. On the other hand, it also allows to use numerical quadrature methods to approximate higher-order Fréchet derivatives. We demonstrate that in certain situations—in particular when the derivative order k is moderate and the direction terms in the derivative have low-rank structure—the resulting algorithm can outperform established methods from the literature by a large margin.

Inequalities for generalized matrix function and inner product

We present inequalities related to generalized matrix function for positive semidefinite block matrices. We introduce partial generalized matrix functions corresponding to partial traces, and then provide a unified extension of the recent inequalities due to Lin [Electron. J. Linear Algebra 27 (2014) 821-826], Zhang et al. [Linear Algebra Appl. 498 (2016) 99-105] and [Electron. J. Linear Algebra 27 (2014) 332-341] and a result of Choi [Linear Algebra Appl. 532 (2017) 1-7]. Moreover, we demonstrate the application of a positive semidefinite $3\times 3$ block matrix, which motivates us to give alternative proofs of Dragomir's inequality and Krein's inequality.

Matrix-Valued hypergeometric Appell-Type polynomials

<abstract><p>In recent years, much attention has been paid to the role of special matrix polynomials of a real or complex variable in mathematical physics, especially in boundary value problems. In this article, we define a new type of matrix-valued polynomials, called the first Appell matrix polynomial of two complex variables. The properties of the newly definite matrix polynomial involving, generating matrix functions, recurrence relations, Rodrigues' type formula and integral representation are investigated. Further, relevant connections between the first Appell matrix polynomial and various matrix functions are reported. The current study may open the door for further investigations concerning the practical applications of matrix polynomials associated with a system of differential equations.</p></abstract>

Decoupling of second order systems via linear time invariant transformations

Decoupling of linear time invariant (LTI) systems of second order ordinary differential equations (ODEs) with real matrix coefficients via LTI transformations in the configuration space is revisited. Necessary and sufficient conditions for the existence of such transformations are presented, as well as methods to construct transformation matrices. Matrix series and general matrix function formulas for the system’s matrix coefficients which enable decoupling are also discussed. Systems with non-symmetric and symmetric matrices are considered. Situations when systems degenerate into second order ODEs and lower order ODEs, including differential algebraic equations (DAEs), are analyzed as well.

Mathematical Modeling and Functional Order Pursuits with Separated Dynamics

This work is devoted to the study of the pursuit problem in controlled systems described by a fractional-order equation with divided dynamics. For fixed player controls, representations of solutions are established in the form of analogs of the Cauchy formula using generalized matrix functions. Sufficient conditions are obtained for the possibility of completing the pursuit. Specific types of fractional differential equations and models of fractional dynamical systems are considered. The qualitative dynamics, issues of stability and controllability of such systems are discussed. Considered, try which, the motion of the equation is described with irrational orders. Problems of the type under study are encountered in modeling the processes of economic growth and in problems of stabilizing dynamic systems.

Analytical properties of the two-variables Jacobi matrix polynomials with applications

Abstract In the current study, we introduce the two-variable analogue of Jacobi matrix polynomials. Some properties of these polynomials such as generating matrix functions, a Rodrigue-type formula and recurrence relations are also derived. Furthermore, some relationships and applications are reported.

Analytical Properties of the Generalized Heat Matrix Polynomials Associated with Fractional Calculus

In this paper, we introduce a matrix version of the generalized heat polynomials. Some analytic properties of the generalized heat matrix polynomials are obtained including generating matrix functions, finite sums, and Laplace integral transforms. In addition, further properties are investigated using fractional calculus operators.

Continuity, differentiability and semismoothness of generalized tensor functions

<p style='text-indent:20px;'>A large number of real-world problems can be transformed into mathematical problems by means of third-order real tensors. Recently, as an extension of the generalized matrix function, the generalized tensor function over the third-order real tensor space was introduced with the aid of a scalar function based on the T-product for third-order tensors and the tensor singular value decomposition; and some useful algebraic properties of the function were investigated. In this paper, we show that the generalized tensor function can inherit a lot of good properties from the associated scalar function, including continuity, directional differentiability, Fréchet differentiability, Lipschitz continuity and semismoothness. These properties provide an important theoretical basis for the studies of various mathematical problems with generalized tensor functions, and particularly, for the studies of tensor optimization problems with generalized tensor functions.</p>

A General Matrix Function Dimensionality Reduction Framework and Extension for Manifold Learning.

Many dimensionality reduction methods in the manifold learning field have the so-called small-sample-size (SSS) problem. Starting from solving the SSS problem, we first summarize the existing dimensionality reduction methods and construct a unified criterion function of these methods. Then, combining the unified criterion with the matrix function, we propose a general matrix function dimensionality reduction framework. This framework is configurable, that is, one can select suitable functions to construct such a matrix transformation framework, and then a series of new dimensionality reduction methods can be derived from this framework. In this article, we discuss how to choose suitable functions from two aspects: 1) solving the SSS problem and 2) improving pattern classification ability. As an extension, with the inverse hyperbolic tangent function and linear function, we propose a new matrix function dimensionality reduction framework. Compared with the existing methods to solve the SSS problem, these new methods can obtain better pattern classification ability and have less computational complexity. The experimental results on handwritten digit, letters databases, and two face databases show the superiority of the new methods.

Matrix Poincaré inequalities and concentration

We show that any probability measure satisfying a Matrix Poincaré inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carré du champ operator. This extends to the matrix setting a classical phenomenon in the scalar case. Moreover, the proof gives rise to new matrix trace inequalities which could be of independent interest. We then apply this general fact by establishing matrix Poincaré inequalities to derive matrix concentration inequalities for Gaussian measures, product measures and for Strong Rayleigh measures. The latter represents the first instance of matrix concentration for general matrix functions of negatively dependent random variables.

On the Extended Hypergeometric Matrix Functions and Their Applications for the Derivatives of the Extended Jacobi Matrix Polynomial

In this paper, we obtain some generating matrix functions and integral representations for the extended Gauss hypergeometric matrix function EGHMF and their special cases are also given. Furthermore, a specific application for the extended Gauss hypergeometric matrix function which includes Jacobi matrix polynomials is constructed.

Дифференциальные игры преследования дробного порядка с нелинейными управлениями

The article is devoted to the problems of extending the results and methods of the theory of differential games and optimal control to systems of fractional order. The research is motivated by numerous applications of fractional calculus in control problems of industrial facilities, chemical and biochemical plants, etc. The article considers the problem of pursuit in games represented by nonlinear differential equations of arbitrary fractional order in the sense of Caputo. To study this pursuit problem, we use an approach similar to the method of L. S. Pontryagin, developed for linear differential games of integer orders. In this paper, new sufficient conditions are obtained for solving the pursuit problem in the class of games under study. It has been proven that if these conditions are met, the game can be completed within a certain limited period of time. When solving the pursuit problem, we also used the representation of the solution to a differential equation in terms of generalized matrix functions.

Certain properties of Konhauser matrix polynomials via Lie Algebra techniques

The principal purpose of this paper is devoted to an investigation of some interesting generating matrix functions for the second-kind Konhauser matrix polynomials (KMPs) using a Lie group theory. We derive many interesting properties such as Rodrigues formula, integral representations, matrix recurrence relations, matrix differential equation, finite sums and generating matrix functions for the second-kind KMPs.