Ciro Santilli If is the change of basis matrix, then the matrix representation of a bilinear form 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.
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: