Let T e an invertible linear operator. Prove that a scalar Lamda is an eigenvalue of T. if and only if the inverse of lamda is and eigen value of T^-1.
Could it be this simple?:
$\displaystyle T^{-1} v = \frac{1}{\lambda} v \Rightarrow T \, T^{-1} v = \frac{1}{\lambda} T \, v \Rightarrow \lambda v = T \, v$.
$\displaystyle T \, w = \lambda w \Rightarrow T^{-1} T \, w = T^{-1}\lambda \, w \Rightarrow \frac{1}{\lambda} \, w = T^{-1} w$.
Details and justifications left to you.