Rank restricting functions

In this paper we characterize, for given positive integers k and d, the class of functions such that for every n×m real-valued matrix A=(aij)i=1nj=1m (arbitrary n and m) of rank at most k, the matrix f(A)=(f(aij))i=1nj=1m has rank at most d, as well as the class of functions such that for every n×m complex-valued matrix A=(aij)i=1nj=1m (arbitrary n and m) of rank at most k, the matrix g(A)=(g(aij))i=1nj=1m has rank at most d. For k 2 each such function f is a polynomial of an appropriate form which we shall exactly delineate, while each g is a polynomial in z and , also of an explicitly delineated form. For k=1 the class of such functions, in each case, is significantly different. Nonetheless it is via the study of the case k=1 that we are able to characterize such functions where k 2.