The Upper Triangular Matrix Guarantees The Induced Mapping Of The Invariants

The study of Preserving map about all kinds of invariants in matrix space have been the concern in many fields.Suppose F is a field and n>2 is an integer.Denote the set of all n×n upper triangular matrices over F by Tn(F).Let fij,jE[1,n])be a function of F,where[1,n]denotes the set {1,2 …n}.If f is defined by f:A =(aij)?(fij(?ij)),(?)A?Tn-(F),we say f is induced by {fij|i,j?[1,n]}.Suppose f is induced mapping by fij on Tn(F),if A and B is similarity,then f(A)and f(B)is also similarity,and we say f is preserving similarity;If A + B = C,is rank A + rankB = ran,kC = rank(A + B),then we say A,B is rank-additivity.If A,B is rank-additivity means f(A),f(B)is rank-additivity namely rank f(A)+rankf(B)= rank(f(A)+ f(B)),then we say f is preserving rank-additivity.In this paper,we characterize induced maps of preserving similarity and rank-additivity of upper triangular matrix over fields,respectively,We also characterize induced maps of preserving involutory matrix over skew fields.