Linear operators on matrices: The invariance of rank-k matrices

AbstractLet T be a linear operator on the space of all m × n complex matrices such that the rank of T(A) is the rank of A whenever the rank of A is k. We show that there are nonsingular m × m and n × n matrices U and V respectively such that either T(A) =UAV for all m × n matrices A, or m =n and T(A) =UAtV for all m ×m matrices A