Analogous to a linear form, a bilinear form is a Bilinear map where the image is the underlying field of the vector space, e.g. $R_{n}×R_{m}→R$.

Some definitions require both of the input spaces to be the same, e.g. $R_{n}×R_{n}→R$, but it doesn't make much different in general.

The most important example of a bilinear form is the dot product. It is only defined if both the input spaces are the same.

As usual, it is useful to think about how a bilinear form looks like in terms of vectors and matrices.

Unlike a linear form, which was a vector, because it has two inputs, the bilinear form is represented by a matrix $M$ which encodes the value for each possible pair of basis vectors.

If $C$ is the change of basis matrix, then the matrix representation of a bilinear form $M$ that looked like:
then the matrix in the new basis is:
Sylvester's law of inertia then tells us that the number of positive, negative and 0 eigenvalues of both of those matrices is the same.

$B(x,y)=x_{T}My$

$C_{T}MC$

Proof: the value of a given bilinear form cannot change due to a change of bases, since the bilinear form is just a function, and does not depend on the choice of basis. The only thing that change is the matrix representation of the form. Therefore, we must have:
and in the new basis:
and so since:

$x_{T}My=x_{new}M_{new}y_{new}$

$x=Cx_{new}y=Cy_{new}x_{new}M_{new}y_{new}=x_{T}My=(Cx_{new})_{T}M(Cy_{new})=x_{new}(C_{T}MC)y_{new}$

$∀x_{new},y_{new}x_{new}M_{new}y_{new}=x_{new}(C_{T}MC)y_{new}⟹M_{new}=C_{T}MC$