Quadratic Form & Bilinear Functions

Sylvester figured all this stuff out in 1852.

Let V be an n-dimensional vector space over the reals. A quadratic form is a function q(v) from vectors to reals—from V to R.
q: V→R

Concretely: (in the following i, j and k take on exactly n distinct values.)

For any quadratic form q on V and any basis e_i for V, there are n² reals g_ij such that
g_ij = g_ji and for any set {α¹, ... αⁿ} ⊂ R, q(Σα^ke_k) = Σ_iΣ_jg_ijαⁱα^j

The g’s determine q and vice versa.
f(Σ_kα^ke_k, Σ_kβ^ke_k) = Σ_iΣ_jg_ijαⁱβ^j is a symmetric bilinear function.
The quadratic form determines the bilinear function and conversely.

If g_ij is a metric then the corresponding bilinear function is the corresponding inner product and conversely.

Abstractly:

a function q from V to R is a quadratic form when there is a symmetric bilinear function f(x, y) to the reals and q(x + y) = f(x, x) + 2f(x, y) + f(y, y).

Every bilinear function f determines a quadratic form q(x) = f(x, x).

Given a quadratic form q there is a basis e_k for which
q(Σα^ke_k) = Σσ^kα_k² and for each k, σ^k ∈ {–1, 0, 1} and σ^k ≥ σ^k+1. There may be more than one such basis but the quadratic form uniquely determines the σ’s, collectively the signature of the form.

Given two quadratic forms, each for their own n-dimensional vector space, there is an isomorphism between the two iff they have the same signature.

See Sylvester’s law

Vector Spaces Without Quadratic Forms?

Once you have learned about quadratic forms, or equivalently pseudo metrics, it may seem that any applied vector space has such a form naturally associated with it. Perhaps, depending on your meaning of ‘naturally’. But consider the temperatures at the zones described here. The available tools used to argue about the stability of these equations treat these temperatures as the components of a vector in its own vector space. The linear transformation, A, of that space which controls the progress of the temperatures, is not symmetric after you include varying heat capacities of the zones. The sum of the squares of the temperatures is not interesting because of varying heat capacities. Still A has eigenvalues that tell you about the stability of the calculations. I recall that the temperatures times the square roots of the heat capacities, taken as coordinates results in a symmetric A. This was not known when the stability results were derived. It remains to be shown that you can always find an appropriate quadratic form.

There is little difference between a pseudo-metric with signature x and another with signature -x; the geometry of {– + + +} is the same as the geometry of {+ – – –}. Their respective Clifford algebras are not isomorphic, however. What gives?