self-adjoint / orthonormal basis of eigenvectors

Question

Let $T$ be a linear operator on a finite dimensional real inner product space $V$.Then $T$ is self-adjoint iff there exists an orthonormal $\beta$ for $V$ consisting of eigenvectors of $T$.

Please provide the proof of this theorem.

Answer 1

This is referred to as the Second Spectral theorem. For a complete proof of both the first and second, look here. I will simply post the part that interests you here as an answer and since you are familiar with the first part and Schur's Theorem you should be able to follow the proof. Of course, in the link provided, the treatment of the subject is complete.

Assume $F = \Bbb{R}$ and $T$ is a self-adjoint linear operator on $V$ .
Then, the characteristic polynomial of $T$ splits, and Schur’s theorem implies that there exists an orthonormal basis $β$ for $V$ such that $A = [T]_β$ is upper-triangular.

The same proof as for the first spectral theorem now works since $T$ is normal, but it is easier to note that since $T^∗=T$, we know that both $A$ and $A^T = A^∗ = A$ are upper triangular. Therefore, $A$ is diagonal.