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Matrix decompositions play a prominent role in the theoretical study and numerical calculation of split quaternion mechanics. This paper, by means of a complex representation of split quaternion matrices, introduces Gaussian elimination of split quaternion matrices, and obtains a complex structure-preserving algorithm for split quaternion matrix LDU decomposition. Numerical examples show that the complex structure-preserving algorithm is more efficient.

This paper aims to present, in a unified manner, the algebraic techniques of eigen-problem which are valid on both the quaternions and split quaternions. This paper studies eigenvalues and eigenvectors of the v-quaternion matrices by means of the complex representation of the v-quaternion matrices, and derives an algebraic technique to find the eigenvalues and eigenvectors of v-quaternion matrices. This paper also gives a unification of algebraic techniques for eigenvalues and eigenvectors in quaternionic and split quaternionic mechanics.

This paper aims to present, in a unified manner, algebraic techniques for linear equations which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, studies the problem of v-quaternionic linear equations by means of a complex representation and a real representation of v-quaternion matrices, and gives two algebraic methods for solving v-quaternionic linear equations. This paper also gives a unification of algebraic techniques for quaternionic and split quaternionic linear equations in quaternionic and split quaternionic mechanics.

In this paper, we focus on discussing diagonal solutions and general solutions of second-order matrix polynomial equation of high degree in complex field. By characterizing some algebraic properties of the mentioned two types of the solutions, we present sufficient conditions that a general second-order matrix polynomial equation has diagonal solutions or general solutions. Analytic expressions of the solutions, as well as the corresponding algorithms for finding the solutions are provided. An example is given so as to verify the theoretical results we have derived.

This paper, by means of complex representation of a split quaternion matrix, studies the problems of right split quaternion eigenvalues and eigenvectors of a split quaternion matrix, and derives algebraic techniques for the right split quaternion eigenvalues and eigenvectors of the split quaternion matrix in split quaternionic mechanics.

The split quaternionic Schrodinger equation ∂ ∂ t | f 〉 = − A | f 〉 plays an important role in split quaternionic mechanics, in which A a split quaternion matrix. This paper, by means of a real representation of split quaternion matrices, studies problems of split quaternionic Schrodinger equation, and gives an algebraic technique for the split quaternionic Schrodinger equation. This paper also derives an algebraic technique for finding eigenvalues and eigenvectors of a split quaternion matrix in split quaternionic mechanics.

In this article, we propose a new method for the numerical solution of a kind of coefficient inverse problem, in which the time dependent leading coefficient and the right hand side in a parabolic equation need to be identified simultaneously, based on additional information of the solutions at interior points in the computational domain. To solve the nonlinear problem, we use linearized second-order scheme for linearizing the quadratic nonlinear term in time and finite element approximation is used in the space. Based on a special decomposition, at new time level, the original problem is transformed into three standard elliptic problems. The results of numerical experiments are presented to confirm the capability and efficiency of the proposed computational algorithm.

The total least squares (TLS) is a method of fitting that is appropriate when there are errors in both the observation vector b = b m × 1 and the data matrix A = A m × n . In this paper, we study the quaternion total least squares (QTLS) problem by means of real representations of quaternion matrices, and derive an algebraic technique for finding solutions of the QTLS problem in quaternionic quantum theory.

A split quaternionic least squares problem is one method of solving overdetermined sets of split quaternion linear equations AX ? B that is appropriate when there is error in the matrix B. In this paper, by means of complex representation and real representation of a split quaternion matrix, we study the split quaternionic least squares problem, derive two algebraic methods for finding solutions of the problems in split quaternionic mechanics.

This paper aims to present, in a unified manner, Cramer’s rule which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, studies Cramer’s rule for the system of v-quaternionic linear equations by means of a complex matrix representation of v-quaternion matrices, and gives an algebraic technique for solving the system of v-quaternionic linear equations. This paper also gives a unification of algebraic techniques for Cramer’s rule in quaternionic and split quaternionic mechanics.