Some Properties Of The Function Matrix Space Security Domain

let F be a field, and n is an integer.Let f be functions from F to itself, Denote the matrix [f(aij)] by Af for any matrix A=[aij] of order n. A function f:Fâ†'F preserves singularity of matrices if A is singular (?) Af is singular and f:Fâ†'F preserves invertibility matrices if A is invertible (?) Af is invertible. We say that in both directions preserving invertibility if it preserves singularity and invertibility for every matrix of order n. The classical adjoint of A, denote by adjA, f is a function preserving classical adjoint of matrices which means f satisfies (ad/A)f=adjAf The map f:Fâ†'F preserves rank additivity,i.e., rank(A)+rank(B)=rank(A+B)(?) rank(Af)+rank(Bf)=rank[(A+B)f] Preserver problems is a hot field in matrix theory, and it has important application value. This article will takes special matrices ways to show functions forms of preserving invertibility,Preserving classical adjoint and rank additivity in matrices space over field.