Given a characteristic polynomial $P$ of matrix $A$ I need to show that the characteristic polynomial $O$ of $A^2$ can't have more different real roots than $P$.
I know that the characteristic polynomial for both cases can be calculated like this:
$P = |A - \lambda I| = 0$
$O = |A^2 - \lambda^2 I| = 0$
But in a general case with $n*n$ matrices they become way too complicated.
Can anyone guide me in the right direction?