Paul. Real Quaternion Conjugate Matrix Space Cubic Linear Operator

Suppose R and Q are the field of all real numbers and the real quaternion division ring respectively. n is an arbitrary positive integer. Let Mn(Q) and SCn(Q) be R- linear space of all n x n matrices and the set of all n x n self-conjugate matrices over Q respectively. Recently, the study of linear (additive) preserve problems has been active subject in matrix theory. And real quaternion self-conjugate matrix play a very important part in matrix theory because of its particularity.At the same time, the use of real quaternion self-conjugate matrix in geostatics and gyroscopic technology becomes more and more important and extensive. All this makes the study of preserve problems of real quaternion self-conjugate matrix be more and more important. Real quaternion division algebra is a kind of finite division algebra over real number field. When all the studys of idempotence preservers, tripotence preservers on matrices over real number field and matrices over complex field and idempotence preservers on SCn(Q) have acquired commendable achievements, the study of tripotence preservers on SCn(Q) comes to mind naturally. Thus, this paper comes into being because of the demand.In this paper, I study the problem in two ways:searching particular matrices and using already known preserving linear operators to study unknown preserving linear operators. First, in the chapter 2, the forms of the tripotence preserving linear operators from SCn(Q) to Mn(Q) are characterized by two methods. Then, in the chapter 3, the forms of inverse preserving linear operators from SCn(Q) to Mn(Q) are characterized by using the similar methods of chapter 2.As a deduction, the forms of the group inverse preserving linear maps from SCn(Q) to Mn(Q) are also given.