Linear maps preserving Drazin inverses of matrices over fields (Q1763829)

For a matrix \(A\in M_{n}(F)\) a matrix \(A^D\in M_{n}(F)\) is called the Drazin inverse of \(A\) if \(A^D\) is a solution of the equations: \[ AX=XA,\;XAX=X, \;A^kXA=A^k . \] The characterization of linear maps \(T\) from \(M_{n}(F)\) to \(M_{m}(F)\)preserving Drazin inverses of matrices (namely, \(T(A^D)=(T(A))^D\)) is obtained. Here \(F\) is constrained to be a field with at least 5 elements of characteristic different from 2. In the case \(char(F)=2\) and \(m=n\) the author described the linear bijections which preserves the Drazin inverses of matrices.