Effect of a change of basis on the matrix of a bilinear form ( $B_{2} = C^{T} B C$ )

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

(14)

then the matrix in the new basis is:

C^{T} M C

(15)

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: