Quadratic transformations on matrices: Rank preservers

Abstract

Let F be an algebraically closed field of characteristic not 2, and let X = (Xij) be the n × n matrix whose entries Xij are independent indeterminates over F. Now let Q(X) = (qij(X)) be another n × n matrix each of whose entries qij(X) is a quadratic F-polynomial in the Xij. The main result in this paper is: for n ≥ 5, Q(X) satisfies rank(A2) = r implies rank(Q(A)) = r, for all A ∈ Fn×n for r = 0, 1, and 2, if and only if there exist invertible matrices P1, P2 in Fn×n such that either Q(X) = P1X2P2 or Q(X) = P1(X2)tP2. © 1996 Academic Press, Inc.