basic of Jordan Form proof

Question

while studying matrix calculus, I face this problem from "Matrix Methods an Introduction". I think this problem from 5.4 of this book gives a good understanding of "Jordan Forms". So I'll be grateful helping me solving it.

Show that if λ is an eigenvalue of A, then it is also an eigenvalue for $S^{-1}AS$ for any nonsingular matrix S.

Answer 1

Hint

If $\ v\ $ is an eigenvector of $\ A\ $ corresponding to the eigenvalue $\ \lambda\ $, what happens when you multiply $\ S^{-1}v\ $ on the left by $\ S^{-1}AS\ $?