A real, nondegenerate n×n symmetric matrix A, and its corresponding symmetric bilinear form Q(v, w) = v^T A w, has signature (p, q) if there is a nondegenerate matrix C such that C^T A C is a diagonal matrix with p 1s and q -1s. In this case, Q(C v, C w) is a diagonal quadratic form. For example, A = [1 | 0 | 0 | 0 0 | 1 | 0 | 0 0 | 0 | 1 | 0 0 | 0 | 0 | -1] gives a symmetric bilinear form Q called the Lorentzian inner product, which has signature (3, 1).

diagonal quadratic form | orthogonal group | quadratic form | symmetric bilinear form | vector space