If A = (a_(i j)) is a diagonal matrix, then Q(v) = v^T A v = sum a_(i i) v_i^2 is a diagonal quadratic form, and Q(v, w) = v^T A w is its associated diagonal symmetric bilinear form. For a general symmetric matrix A, a symmetric bilinear form Q may be diagonalized by a nondegenerate n×n matrix C such that Q(C v, C w) is a diagonal form. That is, C^T A C is a diagonal matrix. Note that C may not be an orthogonal matrix. For example, consider A = [1 | 2 2 | 3].

diagonal matrix | matrix signature | quadratic form | symmetric bilinear form | vector space