In class, we were told:
let V be a finite dimensional inner product space. A linear map $T: V \to V$ is self if $T = T^*$
Later, we were told:
let V be a finite dimensional real inner prod. space. Let $T: V \to V$ be a LM then T has an orthonormal eigenbasis iff T is self-adjoint
However (my professor has not done any proofs lately), I am a bit confused. Suppose you have a non-orthonormal eigenbasis, and you have a diagonal standard matrix $D$.
Would $D^*$ not be equal to $D$, so the condition of the eigenbasis being orthonormal is violated?
What am I missing? Proof-based insight is appreciated.