Commutative Matrices Research Articles

Algebraic method for LU decomposition in commutative quaternion based on semi‐tensor product of matrices and application to strict image authentication

As a kind of commutative and associative four‐dimensional algebra, commutative quaternion has better applications in color image and signal processing than quaternion. Matrix decomposition is of great concern in the theoretical study and numerical calculation of commutative quaternion. Two kinds of disadvantages of commutative quaternion make the decomposition of commutative quaternion matrix extremely difficult. On one hand, commutative quaternion is not a kind of complete four dimensional division algebra because of the zero divisors. On the other hand, computing the inverse of commutative quaternion is very complicated. In this paper, the semi‐tensor product (STP) of matrices will be used to overcome the above two kinds of shortcomings. And we will propose a real structure‐preserving algorithm based on STP of matrices for commutative quaternion LU decomposition, which makes full use of high‐level operations, relation of operations between commutative quaternion matrices and their ‐representation matrices. Numerical experiments will be provided to demonstrate the efficiency of the real structure‐preserving algorithm based on STP of matrices. Meanwhile, we will apply the structure‐preserving algorithm for strict image authentication.

TWO ALGEBRAIC ALGORITHMS FOR THE LU DECOMPOSITION OF COMMUTATIVE QUATERNION MATRICES AND THEIR APPLICATIONS

The paper proposes two algebraic algorithms for the LU decomposition of com- mutative quaternion matrices. The first of them is an algebraic algorithm based on a complex representation matrix, and the second is a complex structure-preserving algorithm based on Gaussian elimination of commutative quaternion matrices. With their help, it was established that the LU decomposition of commutative quaternion matrices is not unique. The paper also presents the results of numerical implementations of a mathematical model for color image restoration using the proposed algorithms, which demonstrated the higher efficiency of the first algorithm.

The Transformation Matrices of vec A to vec A^T for Diagonal Matrix

This study aims to discusses the transformation matrices of vec A to vec A^T for diagonal matrix. The matrix used is a 4 x 4 diagonal matrix assuming that two pairs of elements are located on the main diagonal with the same value. To get the transformation matrices, this uses the commutation matrix equation is K_mn vec(A)=vec(A^T ). Therefore, several K_mn commutation matrices are obtained on a 4 x 4 diagonal matrix.

The Weak Commutation Matrices of Matrix with Duplicate Entries in its Secondary Diagonal

The Weak Commutation Matrices of Matrix with Duplicate Entries in its Secondary Diagonal

On log-sum inequalities

ABSTRACT In information theory, the well-known log-sum inequality is a fundamental tool which indicates the non-negativity for the relative entropy. In this article, we establish a set of inequalities which are similar to the log-sum inequality involving two functions defined on scalars. The parametric extended log-sum inequalities are shown. We extend these inequalities for the commutative matrices. In addition, utilizing the Löwner partial order relation and the Hansen–Pedersen theory for non-commutative positive semi-definite matrices we demonstrate a number of matrix-inequalities analogous to the log-sum inequality.

TrES: Tropical Encryption Scheme Based on Double Key Exchange

Shor’s quantum algorithm establishes a polynomial time attack on the discrete logarithm problem in any group. Recent announcements of progress in building quantum computers highlight the need for new concepts to create cryptosystems that are resistant to quantum attacks. In this paper, we present а new message encryption scheme. To enhance the security of the scheme, we suggest double key-exchange protocol (KEP). The first stage of the key exchange uses a matrix power function (MPF) in a tropical semiring. These functions are based on the action of a matrix semiring acting on some matrix set. MPFs can be considered as one-way functions because they are based on some generalized satisfiability problems that are potentially NP-complete. The obtained shared secret key at the first stage of the key exchange serves as an input for the second stage. The security of the second phase relies on the difficulty of the semiring action problem. In our protocol, we suggest using left or right action of the tropical semiring (which can be both min-plus and max-plus) on the group of commutative matrices (circulant matrices, in our case). The fact that the key-exchange protocol works in two phases contributes to its security, since an attacker needs to solve two difficult problems in order to break it. The main advantages of the presented protocol are: the increased efficiency and improved security.

A Study on Commutative Elliptic Octonion Matrices

Abstract In this study, firstly notions of similarity and consimilarity are given for commutative elliptic octonion matrices. Then the Kalman-Yakubovich s-conjugate equation is solved for the first conjugate of commutative elliptic octonions. Also, the notions of eigenvalue and eigenvector are studied for commutative elliptic octonion matrices. In this regard, the fundamental theorem of algebra and Gershgorin’s Theorem are proved for commutative elliptic octonion matrices. Finally, some examples related to our theorems are provided.

The Commutation Matrices of Elements in Kronecker Quaternion Groups

This article discusses the commutation matrix in the Kronecker quaternion group; that is, a non-abelian group whose 32 elements are matrices of 4 × 4 size, with entries in the set of complex numbers. The purpose of this paper is to describe the commutation matrices obtained in relation to the matrices in this group. The commutation matrix is a permutation matrix that associates the relationship between the vec and vec of the transpose matrix. Based on the classification of matrices in the Kronecker quaternion group, there are 16 classification of commutation matrices for the matrices in this group.

Observability and Geometric Approach of 2D Hybrid Systems

A connection is emphasized between two branches of the Systems Theory, namely the Geometric Approach and 2D Systems, with a special regard to the concept of observability. An algorithm is provided which determines the maximal subspace which is invariant with respect to two commutative matrices and which is included in a given subspace. Observability criteria are obtained for a class of 2D systems by using a suitable 2D observability Gramian and some such criteria are derived for LTI 2D systems, as well as the geometric characterization of the subspace of unobservable states. The presented algorithm is applied to determine this subspace.

Attasi nD Systems and Polynomial System Solving

Firstly it will be shown that, using the concept of monomial orderings, the classical 1D theory for linear systems generalizes in a very natural way to (autonomous) Attasi nD-systems, giving the Attasi-Hankel matrix, the Attasi transfer function and Attasi state-space realizations. Secondly we will explain how one can associate an Attasi system to any set of polynomial equations having a finite number of solutions and how Attasi realization theory can be used to (1) describe a commutative matrix solution to the set of polynomial equations (this generalizes the Cayley-Hamilton theorem) and (2) find the (scalar) solutions to the set of polynomial equations by computing the joint eigenvalues and eigenvectors of the commuting matrices. Some remarks will be made about how the (discrete time) Attasi system equations can be solved recursively and how that can be used in large-scale eigensolvers. Using the associated Attasi system to a set of polynomial equations also helps to understand various different approaches to polynomial equation solving and multivariate polynomial and rational function minimization.

Commutable Matrix-Valued Functions and Operator-Valued Functions

A simple expression is established for an analytic commutable matrix-valued function. Then a characterization of two by two functional commutative matrices is proven. Finally, a family of analytic normal compact operators on a Hilbert space, which commute with their derivatives, is shown to be functionally commutative.

ON RANDOM WEIGHTED SUM OF POSITIVE SEMI-DEFINITE MATRICES

Let $A_1, \dots, A_n$ be fixed positive semi-definite matrices, i.e. $A_i \in \mathbb{S}_p^{+}(\mathbf{R})$ $\forall i \in \{1, \dots, n\}$ and $u_1, \dots, u_n$ are i.i.d. with $u_i \sim \mathcal{N}(1, 1)$. Then, the object of our interest is the following probability
 $$\mathbb{P}\bigg(\sum_{i=1}^n u_i A_i \in \mathbb{S}_p^{+}(\mathbf{R})\bigg).$$
 In this paper we examine this quantity for pairwise commutative matrices. Under some generic assumption about the matrices we prove that the weighted sum is also positive semi-definite with an overwhelming probability. This probability tends to $1$ exponentially fast by the growth of number of matrices $n$ and is a linear function with respect to the matrix dimension $p.$

Right Semi-Tensor Product for Matrices Over a Commutative Semiring

This paper generalizes the right semi-tensor product for real matrices to that for matrices over an arbitrary commutative semiring, and investigates its properties. This product is defined for any pair of matrices satisfying the matching-dimension condition. In particular, the usual matrix product and the scalar multiplication are its special cases. The right semi-tensor product turns out to be an associative bilinear map that is compatible with the transposition and the inversion. The product also satisfies certain identity-like properties and preserves some structural properties of matrices. We can convert between the right semi-tensor product of two matrices and the left semi-tensor product using commutation matrices. Moreover, certain vectorizations of the usual product of matrices can be written in terms of the right semi-tensor product.

Algebraic techniques for least squares problems in commutative quaternionic theory

Due to the rise of commutative quaternion in Hopfield neural networks, digital signal, and image processing, one encounters the approximate solution problems of the commutative quaternion linear equations and . This paper, by means of real representation and complex representation of commutative quaternion matrices, introduces concepts of norms of commutative quaternion matrices and derives two algebraic techniques for finding solutions of least squares problems in commutative quaternionic theory.

Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism

Let $X$ be a projective variety admitting a polarized (or more generally, int-amplified) endomorphism. We show: there are only finitely many contractible extremal rays; and when $X$ is $\mathbb{Q}$-factorial normal, every minimal model program is equivariant relative to the monoid $SEnd(X)$ of all surjective endomorphisms, up to finite index. Further, when $X$ is rationally connected and smooth, we show: there is a finite-index submonoid $G$ of $SEnd(X)$ such that $G$ acts via pullback as diagonal (and hence commutative) matrices on the Neron-Severi group; the full automorphisms group $Aut(X)$ has finitely many connected components; and every amplified endomorphism is int-amplified.

Sufficient conditions for dispersiveness of invariant control affine systems on the Heisenberg group

The notion of dispersiveness in control systems is characterized by the absence of recursive properties. It implies uncontrollability and Poisson instability. The present manuscript exhibits sufficient conditions for an invariant control affine system on the Heisenberg group to be dispersive. For a control affine system with pairwise commutative matrices, the condition for dispersiveness is the drift not depend linearly on the controlled matrices. For a general control affine system, with bounded control range, the condition is the drift not be an affine transformation of the controlled matrices. These results consequently yield necessary conditions for controllability.

Commutation matrices and commutation tensors

ABSTRACT The commutation matrix was first introduced in statistics as a transposition matrix by Murnaghan in 1938. In this paper, we first investigate the commutation matrix which is employed to transform a matrix into its transpose. We then extend the concept of the commutation matrix to commutation tensor and use the commutation tensor to achieve the unification of the two formulae of the linear preserver of the matrix rank, a classical result of Marcus in 1971.

Some Equivalence Relations and Results over the Commutative Quaternions and Their Matrices

Abstract In this paper, we give some equivalence relations and results over the commutative quaternions and their matrices. In this sense, consimilarity, semisimilarity, and consemisimilarity over the commutative quaternion algebra and commutative quaternion matrix algebra are established. Equalities of these equivalence relations are explicitly determined. Also Syvester-s-Conjugate commutative quaternion matrix equations are studied by means of real representation of the commutative quaternion matrices and consimilarity of the two commutative quaternion matrices.

Extended biorthogonal matrix polynomials

The pair of biorthogonal matrix polynomials for commutative matrices were first introduced by Varma and Tasdelen in [22]. The main aim of this paper is to extend the properties of the pair of biorthogonal matrix polynomials of Varma and Tasdelen and certain generating matrix functions, finite series, some matrix recurrence relations, several important properties of matrix differential recurrence relations, biorthogonality relations and matrix differential equation for the pair of biorthogonal matrix polynomials J(A,B) n (x, k) and K(A,B) n (x, k) are discussed. For the matrix polynomials J(A,B) n (x, k), various families of bilinear and bilateral generating matrix functions are constructed in the sequel.

English

The classical Hermite-Hermite matrix polynomials for commutative matrices were first studied by Metwally et al. (2008). Our goal is to derive their basic properties including the orthogonality properties and Rodrigues formula. Furthermore, we define a new polynomial associated with the Hermite-Hermite matrix polynomials and establish the matrix differential equation associated with these polynomials. We give the addition theorems, multiplication theorems and summation formula for the Hermite-Hermite matrix polynomials. Finally, we establish general families and several new results concerning generalized Hermite-Hermite matrix polynomials.