On Solutions Of Two Kinds Of Split Quaternion Matrix Equations

In 1849, James Cockle found split quaternion. All split quaternions construct a ring which contains zero divisors, nilpotent elements and nontrivial idempotents. The ring is a noncommutative Clifford algebra. Split quaternion and split quaternion matrix theories play important roles in the studies of mathematics and physical applications. From the view of mathematics, split quaternion can be regarded as extensions of complex, but it abandons the principle of product commutative; from the view of physics, split quaternion is the basis of modern quantum mechanics, it has close links to complexified classical quantum mechanics and non-Hermitian quantum mechanics. Split quaternion quantum mechanics has become one of the branches of quantum mechanics.In the study of split quaternion and matrix theories, one often meets a problem of solving split quaternion matrix equation. Because of the noncommutative of split quater-nions, complex methods of resolution break down. For this reason, the study of split quaternion matrix equation is more difficult. This paper studies the split quaternion ma-trix equation X - A(?)B = C and A(?) - XB = C. First, we introduce background and research significance of split quaternion and split quaternion matrix equation. Second,we give basic knowledge and algebraic method of split quaternion matrix, especially real representation method. Third, by using real representation method, we turns the problem of X - A(?)B - C into that of a real matrix equation, then by using Kronecter product and vector operator, we present a necessary and sufficient condition for existence of solutions,give method for finding solutions and derive closed-form solutions of equation. Finally,we study the equation A(?) - XB = C by using Roth theorem and characteristic polynomi-al of coefficient matrix, a necessary and sufficient condition for existence of solutions is given, then a method for finding solutions and an application of j consimilarity to the split quaternion matrix are introduced.