Stiefel-Whitney invariants for bilinear forms

We examine potential extensions of the Stiefel-Whitney invariants from quadratic forms to bilinear forms which are not necessarily symmetric. We show that as long as the symbolic nature of the invariants is maintained, some natural extensions carry only low dimensional information. In particular, the generic invariant on upper triangular matrices is equivalent to the dimension and determinant. Along the process, we show that every non-alternating matrix is congruent to an upper triangular matrix, and prove a version of Wittʼs Chain Lemma for upper-triangular bases. (The classical lemma holds for orthogonal bases.)