Don't understand this diagonalizable matrix problem?

To be honest it has been a while since I've taken Linear Algebra. I'm in my differential equations class and do not recall on how to do this problem.

Given: The diagonalizable matrix A. Show that the eigenvalues of A^2 are the

squares of the eigenvalues of A but that A and A^2 have the same eigenvectors.

Re: Don't understand this diagonalizable matrix problem?

Quote:

Originally Posted by

**Reefer** Given: The diagonalizable matrix A. Show that the eigenvalues of A^2 are the squares of the eigenvalues of A but that A and A^2 have the same eigenvectors.

__Hint__ If $\displaystyle Ax=\lambda x\;(x\neq 0)$ then, $\displaystyle A^2x=A(Ax)=A(\lambda x)=\lambda (Ax)=\lambda \lambda x=\lambda^2x.$