Bilinearform

A bilinear form on a vector space V over a field F is a function B: V × V → F that is linear in each argument. That is, for all u, v, w in V and a in F, B(u+v, w) = B(u, w) + B(v, w), B(au, v) = a B(u, v), and similarly linear in the second argument: B(u, v+w) = B(u, v) + B(u, w), B(u, av) = a B(u, v). The study of bilinear forms often emphasizes symmetry, nondegeneracy, and how the form changes under a change of basis.

With respect to a basis, a bilinear form is represented by a matrix A such that B(u, v) = u^T A v, where u and v are the coordinate column vectors of the vectors in V. A is symmetric if B(u, v) = B(v, u) for all u, v; A is symmetric in that case. In general, changing the basis changes A by a congruence transformation A ↦ P^T A P.

Special classes include symmetric bilinear forms, where B(u, v) = B(v, u); and alternating (or skew-symmetric) forms, where B(v, v) = 0 for all v (in characteristic not 2, alternating implies skew-symmetric). A bilinear form is nondegenerate if the only vector v with B(v, w) = 0 for all w is v = 0; equivalently, its Gram matrix has nonzero determinant.

Examples include the standard dot product on R^n, B(u, v) = u^T v, which is symmetric and positive definite; and, on R^2, the alternating form B((x1, y1), (x2, y2)) = x1 y2 − y1 x2, which is skew-symmetric and nondegenerate. Bilinear forms underpin the theory of quadratic forms, classification up to change of basis, and constructions in geometry and physics, including symplectic and metric structures.