Additive Preservers Of Determinant On Self-Conjugate Quaternion Matrix Spaces

Linear Preserver Problems (LPPs) is a very active topic in the field of matrix theory research. The linear operator which characterize the preserving some invariant between matrix sets is called"Linear Preserver Problems". The earliest paper about LPP is Frobenius'linear transformation of preserving determinant. After that, in the 1960s, American matrix theory expert Marcus had studied the core problem of preserving rank one. Since M . Omladic and P . Semel replaced linear operators with additive operators to consider the problems in 1991, the study of Additive Preserver Problems has begun, It is obvious that additive preserver problems is Linear Preserver Problems'extension. In these years, Some mathematicians have done many works in preserver problems, it contains: preserving idempotence, preserving inverse, preserving Moore-Penrose inverse, preserving rank-one, preserving adjugate and so on. The study of preserver determinant is this paper's main research content.For the problems of preserver determinant, there are some good results on full matrix spaces over complex number field, on full matrix spaces over any filed, on upper-triangular matrix spaces over any field and skew-symmetric matrix spaces over any filed. In 2004, Zhang Xian and Cao Chongguang studied the preserver determinant on symmetric matrix spaces over the field.In recent years, with the wide range of development of quaternion matrix on the quantum rigid body's mechanics and statistics, the research of quaternion matrix's theory and computation is developing deeply step by step, the special quaternion matrix, such as self-conjugate quaternion matrix is the important part of quaternion matrix. Based on the different determinant on quaternion division ring Q defined by many mathematicians home and abroad, this paper mainly introduces the determinant and double determinant defined by Prof. Chen Longxuan, for the general quaternion matrix, its computation is complicated.Based on these researches, we mainly introduce the special quaternion matrix--self-conjugate quaternion matrix, we consider the preserver of determinant on field extended to the self-conjugate matrix spaces on non-commutative real quaternion division ring, when n = 2and n≥3, we characterize...