Bilineal forms

Given a vector space V, we will say a function of  f: VxV-> R is bilineal if

f(x,y) = μ  belonging to R and being x,y two vectors.

In order to be a bilineal form, it has to be linear in both positions.

Lineal to the left : f(ax+by,z) = af(x,z) + bf(y,z)
Lineal to the right : f(x, ay+bz)=af(x,y) + b(x,z)

There are two types of bilineal forms:

• Symmetrical: f(x,y) = f(y,x)
• Antisymmetrical: f(x,y) = -f(y,x)

Bilinear forms are used to calculate a real number using the image of two vectors. A particular case of this operation is when the image equals the two vectors multiplied by one (eigenvalue = 1), meaning that we have an euclidean space and therefor, a scalar product. The matrix form works similar to the scalar product. You multiply the x vector as a row, the matrix is the image of the combination of the base (e1,e1; e2,e1; e3,e1; e2,e2; e2,e3; e3,e3) and finally the vector is placed as a column. This matrix form of the bilineal function is called "Gramm Matrix".
In this link you can find more information about bilineal forms.