The equivalence of bilinear forms

Abstract

Let B: I’ x V -+ F be a bilinear form on the finite dimensional vector space V over the field F. No assumptions such as symmetry or skew symmetry are made, although we assume that B is nondegenerate. Another such form C: W x W + F is equivalent to B if there is an isomorphism q~: V -+ W such that C(cpu, TV) = B(u, V) f or all u and z, in V, such a p is called an isometry of I/ and W, or more precisely of B and C. The solution of the equivalence problem for alternating forms is well known, and solutions have also been obtained for symmetric forms, and more generally for hermitian forms and quadratic forms, over special fields. It seems to be less well known that in most cases the equivalence problem for general bilinear forms has also been “solved” by J. Williamson [lo]. His work was extended by G. E. Wall [9] to the case of sesquilinear forms over a division ring. These solutions consist of associating to B first a linear transformation, called the asymmetry of B, and second a sequence b, ,..., b, of symmetric, alternating, and hermitian forms; then B N C if and only if their asymmetries are similar and b, N ci , b, N ca ,..., b, ‘v c, . Thus once one has solved the equivalence problem for the “classical” types of forms over F (and its finite extensions), the general problem of equivalence is also solved. Williamson’s work was motivated originally by probIems in linear systems of differential equations and subsequently by the problem of determining the conjugacy classes, or “normal forms” of elements of the classical groups. The connection between these conjugacy classes and equivalence classes of bilinear forms can be found in [9]. The conjugacy problem has also been worked on more directly by many people (e.g., Springer [7], Zassenhaus [ll], Cikunov [3], Milnor [5]). I p ro p ose to apply techniques similar to those employed in the conjugacy problem, especially by Milnor, to recast the