On the Diagonalization of Quadratic Forms

Introduction An undergraduate course in linear algebra sometimes includes a treatment of the elementary theory of quadratic forms. This is not surprising since a quadratic form (over a field F of characteristic not 2) is essentially the same as a symmetric bilinear form, which is in turn the same as a symmetric matrix. The main theorem for quadratic forms proved in a typical linear algebra course is that any quadratic form q(x,,..., x,,) can be diagonalized, i.e., after a linear change of variables {xl.x,} -x4 {Y. Yn}, q can be written as n Here, the as's are elements of F, some of which may be equal to zero. The rank of q is defined to be the number of nonzero a 's, and (in case F = R) the signature of q is defined to be r s, where r and s are respectively the number of positive and negative ai's. Both the rank and the signature depend only on the isometry class of the quadratic form q (and does not depend on the particular diagonalization taken); see, e.g., [1, ?5.3], [3, Ch. 9]. While most textbooks offer exercises for the diagonalization of quadratic forms in a small number of variables (say n < 5), there seem to be few good examples for the diagonalization of quadratic forms in n variables. In teaching a course in quadratic form theory, I recently came across the following four explicit n-ary forms: