Quadratic Transformations on Matrices: Rank Preservers Original Research Article

LetFbe an algebraically closed field of characteristic not 2, and letX=(Xij) be then×nmatrix whose entriesXijare independent indeterminates overF. Now letQ(X)=(qij(X)) be anothern×nmatrix each of whose entriesqij(X) is a quadraticF-polynomial in theXij. The main result in this paper is: forn≥5,Q(X) satisfies rank(A2)=rimplies rank(Q(A))=r, for allAset membership, variantFn×nforr=0, 1, and 2, if and only if there exist invertible matricesP1, P2inFn×nsuch that eitherQ(X)=P1X2P2orQ(X)=P1(X2)tP2.