Algorithm Research On Several Classes Of Algebra Problems In Split Quaternion Matrices

The set of split quaternions is an associative and noncommutative 4-dimensional Clifford algebra, and it contains zero divisors, nilpotent elements and nontrivial idempotents. Split quaternion ring and quaternion ring are two different noncommutative 4-dimensional Clifford algebra; the latter is a non-commutative skew-field, but the former is not. Therefore, the algebraic structure of split quaternion ring is more complicated than that of quaternion ring.When physicists research the relation between complexified classical and non-Hermitian quantum mechanics, they found that there are surprising links to quaternionic and split quaternionic mechanics. This achievement leads to the possibility of applying algebraic techniques for quaternions and split quaternions to solve some challenging and open issues in complexified classical and quantum mechanics.In this paper, we mainly discuss algebraic techniques for split quaternions in split quater-nionic mechanics. And the problems of solutions of split quaternion matrices equations, the inverse matrix and the rank of split quaternion matrix are studied. The text structure is as follows:In chapter 1, the main content is to introduce the research background and development status of algebraic techniques for split quaternions in split quaternionic mechanics, and the main results of this paper.In chapter 2, the main content is to research the problems of diagonalization of a split quaternion matrix. For this reason, define the complex representation matrix and real rep-resentation matrix of a split quaternion matrix. Then,gives algebraic methods for diagonal-ization of a split quaternion matrix in split quaternionic mechanics. Finally, a corresponding example is used to illustrate the applicability of the proposed method.In chapter 3, the main content is to research algebraic method for solving split quater-nionic linear equations. First, define the complex representation that is different from the question 1 of a split quaternion matrix. Then, introduce a definition of rank of a split quaternion matrix, and an algebraic method and an algorithm for split quaternionic lin-ear equations are obtained in split quaternionic quantum theory. Finally, a corresponding example is used to illustrate the applicability of the proposed method.In chapter 4, the main content is to research Cramer rule for solving the problem of solutions of split quaternionic linear equations. First, by means of complex representation of split quaternion matrix in question 2, we propose new definitions of determinant and adjoint matrix for a split quaternion matrix. Then,derive a technique of finding an inverse matrix of an invertible split quaternion matrix, and the Cramer rule for split quaternionic linear equations is obtained in split quaternionic quantum theory. Finally, a corresponding example is used to illustrate the applicability of the proposed method.In chapter 5, the main content is to research the zeros of split quaternion quadratic polynomial equations(SQQP)x2+bx+c = 0. By the theory of solving the real system of linear equations, the split quaternion quadratic polynomial (SQQP) equation with one unknown is transformed into a parametric system of linear equations with a quadratic constraint.Moreover, an algorithm for solving the SQQP equation with one unknown is obtained in split quaternionic quantum theory, and a corresponding example is used to illustrate the applicability of the proposed method.