= Effect of a change of basis on the matrix of a bilinear form
{title2=$B_2 = C^T B C$}
If $C$ is the <change of basis matrix>, then the <matrix representation of a bilinear form> $M$ that looked like:
$$
B(x,y) = x^T M y
$$
then the matrix in the new basis is:
$$
C^T M C
$$
<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.
Proof: the value of a given bilinear form cannot change due to a \x[change of basis]{magic}, since the bilinear form is just a <function (mathematics)>, 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:
$$
x^T M y = x_{new}^T M_{new} y_{new}
$$
and in the new basis:
$$
x = C x_{new} \\
y = C y_{new} \\
x_{new}^T M_{new} y_{new} = x^T M y = (Cx_{new})^T M (Cy_{new}) = x_{new}^T (C^T M C) y_{new} \\
$$
and so since:
$$
\forall x_{new}, y_{new} x_{new}^T M_{new} y_{new} = x_{new}^T (C^T M C) y_{new} \implies M_{new} = C^T M C \\
$$
Related:
* https://proofwiki.org/wiki/Matrix_of_Bilinear_Form_Under_Change_of_Basis