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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.

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.

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.

This paper studies the constrained least squares problems of quaternion matrices by means of complex representation of quaternion matrices, and derive a novel algebraic technique for the quaternion equality constrained least squares (LSE) and quadratic inequality constrained least squares (LSQI) problems by generalized singular values decomposition (GSVD) of complex matrices.

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.

Quaternion least squares (QLS) problem is one method of solving overdetermined sets of quaternion linear equations $${AX \approx B}$$ and $${AXC \approx B}$$ that are appropriate when there are errors in the matrix B. This paper, by means of complex representation and real representation of a quaternion matrix, studies the QLS problems, and derives two new algebraic techniques for finding solutions of the QLS problem in quaternionic quantum 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.

Coneigenvalues and coneigenvectors are the most important quantities related to consimilarity tranformations of quaternion matrices. By mean of complex representation method, we characterize the relations between principle right coneigenvalues of a quaternion matrix and eigenvalues of corresponding complex representation matrix. In addition, an algorithm based on the complex representation is described for computing the coneigenvalues and coneigenvectors of quaternion matrices. In the end an example is given to illustrate the validity of our algorithm.

In this paper, we introduce a kind of complex representation of quaternion matrices (or quaternion vectors) and quaternion matrix norms, study quaternionic least squares problem with quadratic inequality constraints (LSQI) by means of generalized singular value decomposition of quaternion matrices (GSVD), and derive a practical algorithm for finding solutions of the quaternionic LSQI problem in quaternionic quantum theory.

This paper studies the problem of solution of the quaternion matrix equation $${A \tilde{X} -X B = C}$$ by means of real representation of a quaternion matrix, derives a similarity version of Roth’s theorem for the existence of solution to the quaternion matrix equation, and obtains closed-form solutions of the quaternion matrix equation in explicit forms. As a special case, this paper also gives some applications of the solutions of complex matrix equation $${A \bar{X} -X B = C}$$ and of the consimilarity of complex matrices.

By means of complex representation and real representation of quaternion matrices, paper [7] studied the problem of diagonalization of quaternion matrices. This paper introduces two new complex representation and real representation of quaternion matrices, studies the problem of condiagonalization under consimilarity of quaternion matrices, and gives two algebraic methods for the condiagonalization under consimilarity of quaternion matrices.

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.

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, by means of complex representation of a quaternion matrix, discusses the consimilarity of quaternion matrices, and obtains a relation between consimilarity and similarity of quaternion matrices. It sets up an algebraic bridge between consimilarity and similarity, and turns the theory of consimilarity of quaternion matrices into that of ordinary similarity of complex matrices. This paper also gives algebraic methods for finding coneigenvalues and coneigenvectors of quaternion matrices by means of complex representation of a quaternion matrix.

In this paper, we propose the Method of Particular Solutions for Solving a Class of Nonlinear Reaction-Diffusion Equations using collocation points and approximating the solution using multiquadrics (MQ) and the Thin Plate Splines (TPS) Radial Basis Function (RBF). The scheme works in a similar fashion as finite-difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme.

Two numerical schemes are proposed to solve linear inhomogenenous partial differential equations by using the two-stage MFS-MPS and the one-stage MFS-MPS. The results of numerical experiments are presented. Meanwhile, to research the result, comparing with the results of the two methods and analysing the accuracy of the presented schemes are given by the example.

By means of complex representation of a quaternion matrix, we study the relationship between the solutions of the quaternion equality constrained least squares problem and that of complex equality constrained least squares problem, and obtain a new technique of finding a solution of the quaternion equality constrained least squares problem.