Let φ : V → V {\displaystyle {}\varphi \colon V\rightarrow V} be an isomorphism on a K {\displaystyle {}K} -vector space V {\displaystyle {}V} , and let φ − 1 {\displaystyle {}\varphi ^{-1}} be its inverse mapping. Show that a ∈ K {\displaystyle {}a\in K} is an eigenvalue of φ {\displaystyle {}\varphi } if and only if a − 1 {\displaystyle {}a^{-1}} is an eigenvalue of φ − 1 {\displaystyle {}\varphi ^{-1}} .
