Induced Mapping Of The Guarantee

Research into linear preservers questions is an active area in the matrix and operator algebra, there are lots of research has practical meaning. Suppose that F is a field and n? 2 is an integer. Denote by Mn(F) the set of all n × n matrices over F. Let fij(i,j ? [1,n]) be a function of F, and [1,n] represented by{1,2···n}. If f defined by f:A= (aij)? (fij(aij)),(?)A?Mn(F), We say f induced by{fij|i,,j?[1,n]}.It is easy to see that the mapping is not necessarily a linear or additions. If AA-1=In, means f(A)f(A-1)= In, then we say f is preserving inverses. If A2= In, means (f(A))2= In, then we say f is preserving involutory. In this paper, we characterize induced maps preserving involutory matrices over fields, spread and add some related results about preserving inverse matrices.In this paper, we also characterize induced maps preserving upper triangular involutory matrice over skew fields.