I'd like to know how to use the Gram-Schmidt method to diagonalize a quadratic form.
I tried to look on the internet for this method but I got no results. My teacher told me I could use Gram-Schmidt method to diagonalize a quadratic form, instead of using the eigenvalues method which is much slower. Is there anybody here who knows this method? Please, can you explain me how does it work? I really need it.
Thank you in advance.
Your teacher is clearly wrong. If it would be possible to diagonalize a positive definite quadratic form over $\mathbb{Q}$ with the Gram-Schmidt process only, all the eigenvalues of the associated matrix would be algebraic numbers over $\mathbb{Q}$ with degree $2^m$, given the operations involved in the GS process. If, for instance, the characteristic polynomial of our matrix with coefficients in $\mathbb{Q}$ is an irreducible polynomial with degree $3$, we are doomed.
What actually holds is that we may use the Gram-Schmidt process to find a QR decomposition, but that's a different story, a QR decomposition is not a diagonalization ($Q^{-1} D Q$).
However, we may use the $QR$ algorithm to find an approximate diagonalization. The idea is the following one: we may use GS to write $M$ as $QR$. Then we may consider $M_1=RQ$ and apply GS to $M_1$ to get $M_1=Q_1 R_1$. If we set $M_2=R_1 Q_1$ and continue this way, $R_n$ converges to $D$ and $Q_n$ converges to the matrix $Q$ giving the diagonalization $M=Q^{-1}D Q$.