If T is a normal operator, prove that characteristic vectors for T which are associated with distinct characteristic values are orthogonal.

Question

The question is that If $T$ is a normal operator, prove that characteristic vectors for $T$ which are associated with distinct characteristic values are orthogonal.

my proof is,
let $W_i=$ eigenvector space associated with eigenvalue $c_i$
each $W_i$ is invariant under $T^*$

Would give a hint for this problem?
and
Is is okay with space is not finite?

Related (I'll leave it to someone else to decide if it's a duplicate). — Ben Grossmann, May 26 '17 at 14:08

score 3 · Accepted Answer · answered May 26 '17 at 14:09

Suppose that $T(u)=\lambda u$ and that $T(v)=\mu v$, with $\lambda\neq\mu$; you want to prove that $\langle u,v\rangle=0$. Well,\begin{align*}\lambda\langle u,v\rangle&=\langle T(u),v\rangle\\&=\langle u,T^*(v)\rangle\\&=\langle u,\overline{\mu}v\rangle\\&=\mu\langle u,v\rangle.\end{align*}Since $\lambda\neq\mu$, $\langle u,v\rangle=0$.

If T is a normal operator, prove that characteristic vectors for T which are associated with distinct characteristic values are orthogonal.

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