Perfect pairing

In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately:

The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms.

When K is the field of complex numbers C, one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument.

Let V ≅ Kⁿ be an n-dimensional vector space with basis {e₁, ..., e_n}.

Define the n × n matrix A by A_ij = B(e_i, e_j).

If the n × 1 matrix x represents a vector v with respect to this basis, and analogously, y represents w, then:

Suppose {f₁, ..., f_n} is another basis for V, such that:

where S ∈ GL(n, K).

Now the new matrix representation for the bilinear form is given by: S^TAS.

Every bilinear form B on V defines a pair of linear maps from V to its dual space V^∗. Define B₁, B₂: V → V^∗ by