Bilinear form

In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately:

The dot product on ${\displaystyle \mathbb {R} ^{n}}$ is an example of a bilinear form which is also an inner product. An example of a bilinear form that is not an inner product would be the four-vector product.

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 be an n-dimensional vector space with basis {e₁, …, e_n}.

The n × n matrix A, defined by A_ij = B(e_i, e_j) is called the matrix of the bilinear form on the basis {e₁, …, e_n}.

If the n × 1 matrix x represents a vector x with respect to this basis, and similarly, the n × 1 matrix y represents another vector y, then: $${\displaystyle B(\mathbf {x} ,\mathbf {y} )=\mathbf {x} ^{\textsf {T}}A\mathbf {y} =\sum _{i,j=1}^{n}x_{i}A_{ij}y_{j}.}$$

A bilinear form has different matrices on different bases. However, the matrices of a bilinear form on different bases are all congruent. More precisely, if {f₁, …, f_n} is another basis of V, then $${\displaystyle \mathbf {f} _{j}=\sum _{i=1}^{n}S_{i,j}\mathbf {e} _{i},}$$ where the ${\displaystyle S_{i,j}}$ form an invertible matrix S. Then, the matrix of the bilinear form on the new basis is S^TAS.