Space Matrix. Complex K-idempotent Operator

Let C be the complex field, n be an arbitrary positive integer, we denote by M_n and S_n the space consisting of all n×n matrices over C and the space consisting of all n×n symmetry matrices over C. Let positive integer k≥2, V_n∈{M_n, S_n}, X is called a k－potent matrix if X^k= X, the main result of the paper is to characterize the k－potence preserving mapφ: V_n→M_n, i.e., A－λB is a k－potent matrix in V_n, impliesφ(A)－λφ(B) is a k－potent matrix in M_n, where A, B∈V_n, A∈C.In Chapter 1, we presented a brief introduction to the preserving problems, defined multiplicative, additive, linear and "A－λB" preserving problems, divided preserving problems into four patterns as functions, properties, subsets and transformations according to the property of invariants.The main result of Chapter 2 is, ifφ: M_n→M_n is a k－potence preserving map, then there exit an invertible matrix P∈M_n and c∈C with c^k = c, such thatφ(X) = cPXP^-1 orφ(X) = cPX^tP^-1 for any X∈M_n, where X^t is the transpose of X.In Chapter 3, we proved that if n = 2,φ: S_n→M_n is a k－potence preserving map, then there exit an invertible matrix P∈M_n and c∈C with c^k = c, such thatφ(X) = cPXP^-1 for any X∈S_n.