Assuming that characteristic polynomial $\chi_A$ has roots $\lambda_1, \cdots, \lambda_n$ only of odd power, prove that if $P(\lambda_i) = Q(\lambda_i)$ for all $i \in [1,\cdots,n]$ then $P(A) = Q(A)$.
(This is happening in a vector space over an algebraically closed field).
I don't really have much ideas but I think I have to use Jordan normal form or Caley-Hamilton theorem, I am not sure how to use them tho.
Can you please give me a hint on how to start proving this?
EDIT: it looks like my conjecture is false, sorry for the inconvenience.