Quadratic transformations on matrices: Rank preservers

Let F be an algebraically closed field of characteristic not 2, and let X = (X(ij)) be the n X n matrix whose entries X(ij) are independent indeterminates over F. Now let Q(X) = (q(ij)(X)) be another n X n matrix each of whose entries q(ij)(X) is a quadratic F-polynomial in the X(ij). The main result in this paper is: for n greater than or equal to 5, Q(X) satisfies rank(A(2)) = r implies rank(Q(A)) = r, for all A epsilon F-n X n for r =0, 1, and 2, if and only if there exist invertible matrices P-1, P-2 in F-n X n such that either Q(X) = P-1 X(2) P-2 or Q(X) = P-1(X(2))P-t(2). (C) 1996 Academic Press, Inc.