I am trying to understand these theorems from linear algebra. I never learned this sort of stuff but I think I understand the theorems. Can you tell me if my understanding is correct?

1)Let be a matrix over . If is symmetric then can be diagnolized i.e. written in the form where is a diagnol matrix. But it turns out that not only is diagnolizable it is orthogonally diagnolizable i.e. where is an orthogonal matrix (defined to be such that ). This result is a major result known as the "Spectral Theorem". But then as a consequence of this diagnolization we see that its eigenvectors form an orthogonal basis.

2)How about making #1 more abstract? Let be a finite dimensional inner product space over a field . Let be a linear transformation. We say that is a "symmetric transformation" when its matrix relative to the basis is a symmetric matrix. Then if symmetric its eigenvectors form an orthogonal basis for .