Chapter 6.1, Problem 27E

Let A be an $n\times n$ matrix with an eigenvalue $\lambda$ and let x be an eigenvector belonging to $\lambda .$ Let S be a nonsingular $n\times n$ matrix and let $\alpha$ be a scalar. Show that if
$B=\alpha I-SA{S}^{-1},y=Sx$
then y is an eigenvector of B. Determine the eigenvalue of B corresponding to y?