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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.

The solution of a linear quaternionic least squares (QLS) problem can be transformed into that of a linear least squares (LS) problem with JRS-symmetric real coefficient matrix, which is suitable to be solved by developing structured iterative methods when the coefficient matrix is large and sparse. The main aim of this work is to construct a structured preconditioner to accelerate the LSQR convergence. The preconditioner is based on structure-preserving tridiagonalization to the real counterpart of the coefficient matrix of the normal equation, and the incomplete inverse upper–lower factorization related to only one symmetric positive definite tridiagonal matrix rather than four, so it is reliable and has low storage requirements. The performances of the LSQR algorithm with structured preconditioner are demonstrated by numerical experiments.

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.

First of all, by characterizing solutions of the quaternion equation a x = x ? b , this paper studies consimilarity of quaternions and some related consequences. For the important role of coneigenvalues in consimilarity tranformations of quaternion matrices, this paper further derives the relations between principle right coneigenvalues of a quaternion matrix and eigenvalues of the corresponding real representation matrix. Then, based on the real representation matrix, an effective algorithm is presented to calculate all coneigenvalues and the associated coneigenvectors of a quaternion matrix. Finally, two numerical examples are given to verify the effectiveness of the proposed algorithm.

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.

In the study of theory and numerical computations of quantum theory, in order to well understand the perturbation theory, experimental proposals and theoretical discussions underlying the quaternion and complex formulations, one meets problems of approximate solutions of quaternion and complex problems, such as approximate solution of quaternion and complex linear equations A α ~ ? α λ or A α ? ? α λ that is appropriate when there are errors in the vector α and λ , i.e. quaternion and complex Least Squares coneigen-problem in quantum mechanics. This paper, by means of representation of quaternion matrices and complex matrices, studies the problems of quaternion and complex Least Squares coneigen-problem, and give practical algebraic methods of computing approximate coneigenvalues and coneigenvectors for quaternion and complex matrices in quantum mechanics.

Based on the idea of radial basis functions approximation and the method of particular solutions, we develop in this paper a new meshless computational method to solve nonhomogeneous backward heat conduction problem. To illustrate the effectiveness and accuracy of the proposed method, we solve several benchmark problems in both two- and three-dimensions. Numerical results indicate that this novel approach can achieve an efficient and accurate solution even when the final temperature data is almost undetectable or disturbed with large noises. It has also been shown that the proposed method is stable to recover the unknown initial temperature from scattered final temperature data.

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 Schrodinger equation ∂∂t|f〉=−A|f〉 with A an anti-self-adjoint real quaternion matrix, and |f〉 an eigenstate to A. The quaternionic Schrodinger equation plays an important role in quaternionic quantum mechanics, and it is known that the study of the quaternionic Schrodinger equation is reduced to the study of quaternionic eigen-equation Aα=αλ 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.

Quaternionic least squares (QLS) problem is one method of solving overdetermined sets of quaternion linear equations AX≈B that is appropriate when there is error in the matrix B. In this paper, by means of complex representation of a quaternion matrix, we introduce a concept of norm of quaternion matrices, discuss singular values and generalized inverses of a quaternion matrix, study the QLS problem and derive two algebraic methods for finding solutions of the QLS problem in quaternionic quantum theory.

By means of a complex representation of a quaternion matrix and a companion vector, this paper introduces a new definition of determinant for a quaternion matrix, derives a technique of finding an inverse matrix of a quaternion invertible matrix, and gives a Cramer rule for quaternionic linear equations in quaternionic quantum theory.

By using complex representation and GSVD of quaternion matrices, we define the norm of quaternion matrices, study the equality constrained least squares problem of quaternion matrices, give the necessary and sufficient conditions for the quaternion equality constrained least squares problem to have solutions, and finally derive a practical algorithm.

This paper studies the solutions of complex matrix equations X−AXB=C and X−AXB=C, and obtains explicit solutions of the equations by the method of characteristic polynomial and a method of real representation of a complex matrix respectively.