Real Quaternion Matrix Research Articles

The η-Hermitian solutions to some systems of real quaternion matrix equations

Let Hmxn be the set of all m x n matrices over the real quaternion algebra. We call that A ? Hnxn is ?-Hermitian if A = A?* where A?* = -?A*?,? ? {i,j,k},i,j,k are the quaternion units. In this paper, we derive some solvability conditions and the general solution to a system of real quaternion matrix equations. As an application, we present some necessary and sufficient conditions for the existence of an ?-Hermitian solution to some systems of real quaternion matrix equations. We also give the expressions of the general ?-Hermitian solutions to these systems when they are solvable. Some numerical examples are given to illustrate the results of this paper.

Low Rank Pure Quaternion Approximation for Pure Quaternion Matrices

Quaternion matrices are employed successfully in many color image processing applications. In particular, a pure quaternion matrix can be used to represent red, green, and blue channels of color images. A low-rank approximation for a pure quaternion matrix can be obtained by using the quaternion singular value decomposition. However, this approximation is not optimal in the sense that the resulting low-rank approximation matrix may not be pure quaternion, i.e., the low-rank matrix contains a real component which is not useful for the representation of a color image. The main contribution of this paper is to find an optimal rank-$r$ pure quaternion matrix approximation for a pure quaternion matrix (a color image). Our idea is to use a projection on a low-rank quaternion matrix manifold and a projection on a quaternion matrix with zero real component, and develop an alternating projections algorithm to find such optimal low-rank pure quaternion matrix approximation. The convergence of the projection algorithm can be established by showing that the low-rank quaternion matrix manifold and the zero real component quaternion matrix manifold has a nontrivial intersection point. Numerical examples on synthetic pure quaternion matrices and color images are presented to illustrate the projection algorithm can find optimal low-rank pure quaternion approximation for pure quaternion matrices or color images.

$$\eta $$-Hermitian Solution to a System of Quaternion Matrix Equations

For $$\eta \in \{{\mathbf {i}},{\mathbf {j}},{\mathbf {k}}\}$$ , a real quaternion matrix A is said to be $$\eta $$ -Hermitian if $$A=A^{\eta *},$$ where $$A^{\eta *}=-\eta A^{*}\eta $$ , and $$A^{*}$$ stands for the conjugate transpose of A. In this paper, we present some practical necessary and sufficient conditions for the existence of an $$\eta $$ -Hermitian solution to a system of constrained two-sided coupled real quaternion matrix equations and provide the general $$\eta $$ -Hermitian solution to the system when it is solvable. Moreover, we present an algorithm and a numerical example to illustrate our results.

Some New Properties of The Real Quaternion Matrices and Matlab Applications

In this study, firstly, it was shown that the set of real quaternion matrices is a -dimensional module over the real matrix ring and -dimensional module over the complex matrix ring . Moreover, some new properties of the real quaternion matrices were described. Then, matrix representations of the real quaternion matrices were found easily by Matlab. These matrices were also applied to find the inverse of the real quaternion matrices and inverse matrices were obtained easily with these matrices. In addition, some new properties for matrix representations of the real quaternion matrices were found. Also, the inverse of the real quaternion block matrices was obtained by new methods. Finally, a new method to calculate the determinant of the real quaternion matrices was found and the determinant of these matrices was calculated easily with Matlab application.

Some quaternion matrix equations involving Φ-Hermicity

Let H be the real quaternion algebra and Hmxn denote the set of all m x n matrices over H. For A ? Hm x n, we denote by A? the n x m matrix obtained by applying ? entrywise to the transposed matrix At, where ? is a nonstandard involution of H. A ? Hnxn is said to be ?-Hermitian if A = A?. In this paper, we construct a simultaneous decomposition of four real quaternion matrices with the same row number (A,B,C,D), where A is ?-Hermitian, and B,C,D are general matrices. Using this simultaneous matrix decomposition, we derive necessary and sufficient conditions for the existence of a solution to some real quaternion matrix equations involving ?-Hermicity in terms of ranks of the given real quaternion matrices. We also present the general solutions to these real quaternion matrix equations when they are solvable. Finally some numerical examples are presented to illustrate the results of this paper.

Split‐type octonion matrix

The split and hyperbolic (countercomplex) octonions are eight‐dimensional nonassociative algebras over the real numbers, which are in the form , where em's have different properties for them. The main purpose of this paper is to define the split‐type octonion and its matrix whose inputs are split‐type octonions and give some properties for them by using the real quaternions, split, and hyperbolic (countercomplex) octonions. On the other hand, to make some definitions, we present some operations on the split‐type octonions. Also, we show that every split‐type octonions can be represented by 2 × 2 real quaternion matrix and 4 × 4 complex number matrix. The information about the determinants of these matrix representations is also given. Besides, the main features of split‐type octonion matrix concept are given by using properties of real quaternion matrices. Then, 8n × 8nreal matrix representations of split‐type octonion matrices are shown, and some algebraic structures are examined. Additionally, we introduce real quaternion adjoint matrices of split‐type octonion matrices. Moreover, necessary and sufficient conditions and definitions are given for split‐type octonion matrices to be special split‐type octonion matrices. We describe some special split‐type octonion matrices. Finally, oct‐determinant of split‐type octonion matrices is defined. Definitive and understandable examples of all definitions, theorems, and conclusions were given for a better understanding of all these concepts.

A system of quaternary coupled Sylvester-type real quaternion matrix equations

In this paper we establish some necessary and sufficient solvability conditions for a system of quaternary coupled Sylvester-type real quaternion matrix equations in terms of ranks and generalized inverses of matrices. An expression of the general solution to this system is given when it is solvable. Also, a numerical example is presented to illustrate the main result of this paper. The findings of this paper widely generalize the known results in the literature. The main results are also valid over the real number field and the complex number field.

Simultaneous decomposition of quaternion matrices involving η-Hermicity with applications

Let R and Hm×n stand, respectively, for the real number field and the set of all m × n matrices over the real quaternion algebraH={a0+a1i+a2j+a3k|i2=j2=k2=ijk=−1,a0,a1,a2,a3∈R}.For η ∈ {i, j, k}, a real quaternion matrix A∈Hn×n is said to be η-Hermitian if Aη*=A where Aη*=−ηA*η, and A* stands for the conjugate transpose of A, arising in widely linear modeling. We present a simultaneous decomposition for a set of nine real quaternion matrices involving η-Hermicity with compatible sizes: Ai∈Hpi×ti,Bi∈Hpi×ti+1, and Ci∈Hpi×pi, where Ci are η-Hermitian matrices, (i=1,2,3). As applications of the simultaneous decomposition, we give necessary and sufficient conditions for the existence of an η-Hermitian solution to the system of coupled real quaternion matrix equations AiXiAiη*+BiXi+1Biη*=Ci,(i=1,2,3), and provide an expression of the general η-Hermitian solutions to this system. Moreover, we derive the rank bounds of the general η-Hermitian solutions to the above-mentioned system using ranks of the given matrices Ai, Bi, and Ci as well as the block matrices formed by them. Finally some numerical examples are given to illustrate the results of this paper.

Analogies between random matrix ensembles and the one-component plasma in two-dimensions

The eigenvalue PDF for some well known classes of non-Hermitian random matrices — the complex Ginibre ensemble for example — can be interpreted as the Boltzmann factor for one-component plasma systems in two-dimensional domains. We address this theme in a systematic fashion, identifying the plasma system for the Ginibre ensemble of non-Hermitian Gaussian random matrices G, the spherical ensemble of the product of an inverse Ginibre matrix and a Ginibre matrix G1−1G2, and the ensemble formed by truncating unitary matrices, as well as for products of such matrices. We do this when each has either real, complex or real quaternion elements. One consequence of this analogy is that the leading form of the eigenvalue density follows as a corollary. Another is that the eigenvalue correlations must obey sum rules known to characterise the plasma system, and this leads us to an exhibit of an integral identity satisfied by the two-particle correlation for real quaternion matrices in the neighbourhood of the real axis. Further random matrix ensembles investigated from this viewpoint are self dual non-Hermitian matrices, in which a previous study has related to the one-component plasma system in a disk at inverse temperature β=4, and the ensemble formed by the single row and column of quaternion elements from a member of the circular symplectic ensemble.

A system of generalized Sylvester quaternion matrix equations and its applications

Let Hm×n be the set of all m × n matrices over the real quaternion algebra. We call that A∈Hn×n is η-Hermitian if A=−ηA*η,η ∈ {i, j, k}, where i, j, k are the quaternion units. Denote Aη*=−ηA*η. In this paper, we derive some necessary and sufficient conditions for the solvability to the system of generalized Sylvester real quaternion matrix equations AiXi+YiBi+CiZDi=Ei,(i=1,2), and give an expression of the general solution to the above mentioned system. As applications, we give some solvability conditions and general solution for the generalized Sylvester real quaternion matrix equation A1X+YB1+C1ZD1=E1, where Z is required to be η-Hermitian. We also present some solvability conditions and general solution for the system of real quaternion matrix equations involving η-Hermicity AiXi+(AiXi)η*+BiYBiη*=Ci,(i=1,2), where Y is required to be η-Hermitian. Our results include some well-known results as special cases.

Solution to a system of real quaternion matrix equations encompassing η-Hermicity

Let Hm×n be the set of all m × n matrices over the real quaternion algebra H={c0+c1i+c2j+c3k∣i2=j2=k2=ijk=−1,c0,c1,c2,c3∈R}. A∈Hn×n is known to be η-Hermitian if A=Aη*=−ηA*η,η∈{i,j,k} and A* means the conjugate transpose of A. We mention some necessary and sufficient conditions for the existence of the solution to the system of real quaternion matrix equations including η-HermicityA1X=C1,A2Y=C2,YB2=D2,Y=Yη*,A3Z=C3,ZB3=D3,Z=Zη*,A4X+(A4X)η*+B4YB4η*+C4ZC4η*=D4,and also construct the general solution to the system when it is consistent. The outcome of this paper diversifies some particular results in the literature. Furthermore, we constitute an algorithm and a numerical example to comprehend the approach established in this treatise.

The η-bihermitian solution to a system of real quaternion matrix equations

Let be the set of all matrices over the real quaternion algebra. , , , where is the conjugate of the quaternion . We call that is -Hermitian, if , ; is -bihermitian, if . We in this paper, present the solvability conditions and the general -Hermitian solution to a system of linear real quaternion matrix equations. As an application, we give the necessary and sufficient conditions for the systemto have an -bihermitian solution. We establish an expression of the -bihermitian to the system when it is solvable. We also obtain a criterion for a quaternion matrix to be -bihermitian. Moreover, we provide an algorithm and a numerical example to illustrate the theory developed in this paper.

The exact solution of a system of quaternion matrix equations involving η-Hermicity

A quaternion matrix A is called η-Hermitian if A=Aη∗=-ηA∗η,η∈{i,j,k}, where A∗ is the conjugate transpose of A. In this paper, we consider the following system of linear real quaternion matrix equations involving η-Hermicity:A1X=C1,A2Y=C2,ZB2=D2,A3X+(A3X)η∗+B3YB3η∗+C3ZC3η∗=D3where Y and Z are required to be η-Hermitian. We obtain some necessary and sufficient conditions for the existence of a solution (X,Y,Z) to above system, and give an expression of the general solution when the system is solvable. Our results include the main result of He and Wang (2013) [2] and other well-known results as special cases.

A real quaternion matrix equation with applications

Let i, j, k be the quaternion units and let A be a square real quaternion matrix. A is said to be η-Hermitian if −η A*η = A, where η ∈ {i, j, k} and A* is the conjugate transpose of A. Denote A η* = − η A*η. Following Horn and Zhang's recent research on η-Hermitian matrices (A generalization of the complex AutonneTakagi factorization to quaternion matrices, Linear Multilinear Algebra, DOI:10.1080/03081087.2011.618838), we consider a real quaternion matrix equation involving η-Hermicity, i.e. where Y and Z are required to be η-Hermitian. We provide some necessary and sufficient conditions for the existence of a solution (X, Y, Z) to the equation and present a general solution when the equation is solvable. We also study the minimal ranks of Y and Z satisfying the above equation.

The solvability and the exact solution of a system of real quaternion matrix equations

In this paper, we establish necessary and sucient conditions for the solvability of the system of real quaternion matrix equations 8 > > A1X = C1; Y B1 = D1; A2Z = C2;ZB2 = D2;A3ZB3 = C3; A4X + Y B4 + C4ZD4 = E1: We also present an expression of the general solution to the system. The ndings of this paper widely extend the known results in the literature.

A system of real quaternion matrix equations with applications

Let H be the real quaternion algebra and H n × m denote the set of all n × m matrices over H . Let P ∈ H n × n and Q ∈ H m × m be involutions, i.e., P 2 = I , Q 2 = I . A matrix A ∈ H n × m is said to be ( P , Q ) -symmetric if A = PAQ . This paper studies the system of linear real quaternion matrix equations A 1 X 1 = C 1 X 1 B 1 = C 2 A 2 X 2 = C 3 X 2 B 2 = C 4 A 3 X 1 B 3 + A 4 X 2 B 4 = C c . We present some necessary and sufficient conditions for the existence of a solution to this system and give an expression of the general solution to the system when the solvability conditions are satisfied. As applications, we discuss the necessary and sufficient conditions for the system A a X = C a , XB b = C b , A c XB c = C c to have a ( P , Q ) -symmetric solution. We also show an expression of the ( P , Q ) -symmetric solution to the system when the solvability conditions are met. Moreover, we provide an algorithm and a numerical example to illustrate our results. The findings of this paper extend some known results in the literature.

An algebraic method for Schrödinger equations in quaternionic quantum mechanics

In the study of theory and numerical computations of quaternionic quantum mechanics and quantum chemistry, one of the most important tasks is to solve the Schrödinger equation ∂ ∂ t | f 〉 = − A | f 〉 with A an anti-self-adjoint real quaternion matrix, and | f 〉 an eigenstate to A. The quaternionic Schrödinger equation plays an important role in quaternionic quantum mechanics, and it is known that the study of the quaternionic Schrödinger equation is reduced to the study of quaternionic eigen-equation A α = α λ with A an anti-self-adjoint real quaternion matrix (time-independent). This paper, by means of complex representation of quaternion matrices, introduces concepts of norms of quaternion matrices, studies the problems of quaternionic Least Squares eigenproblem, and give a practical algebraic technique of computing approximate eigenvalues and eigenvectors of a quaternion matrix in quaternionic quantum mechanics.

Complex systems with half-integer spins: Symplectic ensembles

We study the statistical behavior of the Hermitian operators of complex systems with half-integer angular momentum and time-reversal symmetry. The complexity leads to randomization of the operators which can then be modeled, following maximum entropy hypothesis, by multiparametric Gaussian ensembles of real-quaternion matrices. The modeling shows that it is possible to classify the statistical behavior of spin-based complex systems into a continuum of universality classes characterized by a single parameter which is a function of all system parameters.

Distribution and estimation for eigenvalues of real quaternion matrices

In this paper, the concept of generalized spherical neighborhood on the real quaternion field is introduced. Then, Schwartz’ s inequality and the Gerschgorin’s theorem are proved on the real quaternion field and the problems of the distribution and estimation of real quaternion matrix eigenvalues are solved. Finally, some upper bound and lower bound estimation theorems for the moment, the real and imaginary part of the real quaternion matrices eigenvalues are obtained.

Symmetric and skew-antisymmetric solutions to systems of real quaternion matrix equations

In this paper we consider symmetric and skew-antisymmetric solutions to certain matrix equations over the real quaternion algebra H . First, a criterion for a quaternion matrix to be symmetric and skew-antisymmetric is given. Then, necessary and sufficient conditions are obtained for the matrix equation A X = C and the following system { A 1 X = C 1 X B 3 = C 3 to have symmetric and skew-antisymmetric solutions. The expressions of such solutions of the matrix equation and the system mentioned above are also given.