Reference request for quadratic form diagonalization

Question

I want to read a proof of "Every quadratic form q in n variables over a field of characteristic not equal to 2 is equivalent to a diagonal form" using Gram-Schmidt orthogonalization. Could anyone suggets a reference to me?

Answer 1

I am pretty confident that Jacobson's Basic Algebra I covers reducing quadratic forms to diagonal form in chapter 6 Metric vector spaces and the classical groups.

I think I remember another proof, in less generality, in Tensor analysis on manifolds. I remember it was very readable and definitely used Gram-Schmidt.