Let $T:V\rightarrow V$ be a normal operator on a complex space with eigenvalues $c_1,...,c_n$, then there is a polynomial $p$ such that $p(c_i)=\bar c_i$.
I am trying to use the fact that $<v_i,c_iv_i>=\bar c_i$ (assuming $v_i$ is a normal eigenvector for $c_i$). It would be good if I could define $p(c_i)=<v,c_iv>$ for a specifc $v$ but it is not reasonable and I havent used the condition that T is normal.
Hints are appreciated.