A concept of [GRAPHICS] diagonalization matrix over quaternion field is given, the necessary and sufficient conditions for determining whether a quaternion matrix is a [GRAPHICS] diagonalization one are discussed, and...A concept of [GRAPHICS] diagonalization matrix over quaternion field is given, the necessary and sufficient conditions for determining whether a quaternion matrix is a [GRAPHICS] diagonalization one are discussed, and a method of [GRAPHICS] diagonalization of matrices over quaternion field is given.展开更多
Based on the double determinant theory the problem about the determinant of Vandermonde's type over quaternion field is studied, and a necessary and sufficient condition that this double determinant is not equal t...Based on the double determinant theory the problem about the determinant of Vandermonde's type over quaternion field is studied, and a necessary and sufficient condition that this double determinant is not equal to zero is got.展开更多
In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the ta...In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the target structure matrix is constructed by using the complex decomposition of the quaternion matrix, to obtain the necessary and sufficient conditions for the existence of the cyclic solution of the equation and the expression of the general solution. Secondly, the Stein equation is converted into the Sylvester equation by adding the necessary parameters, and the condition for the existence of a cyclic solution and the expression of the equation’s solution are then obtained by using the real decomposition of the quaternion matrix and the Kronecker product of the matrix. At the same time, under the condition that the solution set is non-empty, the optimal approximation solution to the given quaternion circulant matrix is obtained by using the property of Frobenius norm property. Numerical examples are given to verify the correctness of the theoretical results and the feasibility of the proposed method. .展开更多
This paper is an extension of Ref. [1]. Here we study the properties of double determi-nant over quaternion field. The necessary and sufficient existence condition of inverse matrixand its direct expression have been ...This paper is an extension of Ref. [1]. Here we study the properties of double determi-nant over quaternion field. The necessary and sufficient existence condition of inverse matrixand its direct expression have been obtained and the Hadamard theorem has been extendedto the quaternion field.展开更多
A norm of a quaternion matrix is defined. The expressions of the least square solutions of the quaternion matrix equation AX = B and the equation with the constraint condition DX = E are given.
Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. A...Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.展开更多
<p align="justify"> <span style="font-family:Verdana;">It is often claimed that Maxwell’s electromagnetic equations were originally written in terms of quaternions. Once returned to th...<p align="justify"> <span style="font-family:Verdana;">It is often claimed that Maxwell’s electromagnetic equations were originally written in terms of quaternions. Once returned to that form and treated with left and right hand operators as in the mathematics of P. M. Jack, a new seventh scalar electromagnetic field component emerges with possible relations to clean energy extraction and gravitation. It is the purpose here to examine this approach afresh and see how it might link up with other fairly recent, but little known, work in the field. Again, as with the usual form of Maxwell’s equations, a new scalar wave equation is derived but, on this occasion, due to the presence of the scalar component of the quaternion, that equation exhibits a wave speed greater than the speed of light. Historical and present uses within military and humanitarian contexts are considered briefly.</span> </p>展开更多
文摘A concept of [GRAPHICS] diagonalization matrix over quaternion field is given, the necessary and sufficient conditions for determining whether a quaternion matrix is a [GRAPHICS] diagonalization one are discussed, and a method of [GRAPHICS] diagonalization of matrices over quaternion field is given.
文摘Based on the double determinant theory the problem about the determinant of Vandermonde's type over quaternion field is studied, and a necessary and sufficient condition that this double determinant is not equal to zero is got.
文摘In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the target structure matrix is constructed by using the complex decomposition of the quaternion matrix, to obtain the necessary and sufficient conditions for the existence of the cyclic solution of the equation and the expression of the general solution. Secondly, the Stein equation is converted into the Sylvester equation by adding the necessary parameters, and the condition for the existence of a cyclic solution and the expression of the equation’s solution are then obtained by using the real decomposition of the quaternion matrix and the Kronecker product of the matrix. At the same time, under the condition that the solution set is non-empty, the optimal approximation solution to the given quaternion circulant matrix is obtained by using the property of Frobenius norm property. Numerical examples are given to verify the correctness of the theoretical results and the feasibility of the proposed method. .
文摘This paper is an extension of Ref. [1]. Here we study the properties of double determi-nant over quaternion field. The necessary and sufficient existence condition of inverse matrixand its direct expression have been obtained and the Hadamard theorem has been extendedto the quaternion field.
文摘A norm of a quaternion matrix is defined. The expressions of the least square solutions of the quaternion matrix equation AX = B and the equation with the constraint condition DX = E are given.
文摘Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.
文摘<p align="justify"> <span style="font-family:Verdana;">It is often claimed that Maxwell’s electromagnetic equations were originally written in terms of quaternions. Once returned to that form and treated with left and right hand operators as in the mathematics of P. M. Jack, a new seventh scalar electromagnetic field component emerges with possible relations to clean energy extraction and gravitation. It is the purpose here to examine this approach afresh and see how it might link up with other fairly recent, but little known, work in the field. Again, as with the usual form of Maxwell’s equations, a new scalar wave equation is derived but, on this occasion, due to the presence of the scalar component of the quaternion, that equation exhibits a wave speed greater than the speed of light. Historical and present uses within military and humanitarian contexts are considered briefly.</span> </p>
