symmetric bilinear form

A symmetric bilinear form is one where the order of the two inputs does not matter: B(u, v) = B(v, u) for all u, v. The dot product and every inner product are symmetric — swapping the two vectors you are comparing should not change the answer. This single symmetry condition is what makes a form geometrically meaningful, because it lets B measure lengths and angles consistently.

In matrix terms symmetry of the form is exactly symmetry of its matrix: B(u,v) = B(v,u) for all u, v holds if and only if B^T = B. So the symmetric bilinear forms on F^n are in perfect correspondence with the symmetric n-by-n matrices. Congruence preserves this: if B is symmetric then P^T B P is symmetric too, so 'symmetric' is a basis-independent property of the form.

Symmetric forms are the ones that pair with quadratic forms. From any symmetric B you get the quadratic form Q(v) = B(v, v), and conversely polarization recovers B from Q. This back-and-forth means studying symmetric bilinear forms and studying quadratic forms is the same subject seen from two angles — and almost everything below (diagonalization, signature, definiteness) lives in this world.

Honesty note: the equivalence between symmetric forms and quadratic forms needs 2 to be invertible in the field, since polarization divides by 2. Over the reals and complexes that is automatic; over a field of characteristic 2 the theory genuinely splits apart, and many textbooks quietly assume char != 2 throughout this chapter.