Exercise 5.3.4

If $A$ is diagonalizable, we may say that $Q^{-1}\mathit{AQ}=D$ for some invertible matrix $Q$ and for some diagonal matrix $D$ whose diagonal entries are eigenvalues. So $\underset{m\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{D}^m$ exist only when all its eigenvalues are numbers in $S$, which was defined in the paragraphs before Theorem 5.13. If all the eigenvalues are $1$, then the limit $L$ would be $I_n$. If some eigenvalue $\lambda \ne 1$, then its absolute value must be less than $1$ and the limit of ${\lambda }^m$ would shrink to zero. This means that $L$ has rank less than $n$.