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 $\displaystyle A$ be a $\displaystyle n\times n$ matrix over $\displaystyle \mathbb{R}$. If $\displaystyle A$ is symmetric then $\displaystyle A$ can be diagnolized i.e. written in the form $\displaystyle A = MDM^{-1}$ where $\displaystyle D$ is a diagnol matrix. But it turns out that not only $\displaystyle A$ is diagnolizable it is orthogonally diagnolizable i.e. $\displaystyle A = NDN^{-1}$ where $\displaystyle N$ is an orthogonal matrix (defined to be such that $\displaystyle N^{-1} = N^T$). 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 $\displaystyle V$ be a finite dimensional inner product space over a field $\displaystyle F$. Let $\displaystyle T:V\to V$ be a linear transformation. We say that $\displaystyle T$ is a "symmetric transformation" when its matrix relative to the basis is a symmetric matrix. Then if $\displaystyle T$ symmetric its eigenvectors form an orthogonal basis for $\displaystyle V$.