Activity 4.3.4

Let's begin with the diagonal matrix $$D = \left[\begin{array}{rr} 2 & 0 \\ 0 & -1 \\ \end{array}\right]$$. Find the powers $D^2$, $D^3$, and $D^4$. What is $D^k$ for a general value of $k$?

Suppose that $A$ is a matrix with eigenvector $\vvec$ and associated eigenvalue $\lambda$; that is, $A\vvec = \lambda\vvec$. By considering $A^2\vvec$, explain why $\vvec$ is also an eigenvector of $A^2$ with eigenvalue $\lambda^2$.

Suppose that $A= PDP^{-1}$ where $$D = \left[\begin{array}{rr} 2 & 0 \\ 0 & -1 \\ \end{array}\right]$$. Remembering that the columns of $P$ are eigenvectors of $A$, explain why $A^2$ is diagonalizable and find a diagonalization in terms of $P$ and $D$.

Give another explanation of the diagonalizability of $A^2$ by writing $$A^2 = (PDP^{-1})(PDP^{-1}) = PD(P^{-1}P)DP^{-1}$$.

In the same way, find a diagonalization of $A^3$, $A^4$, and $A^k$.

Suppose that $A$ is a diagonalizable $2\times2$ matrix with eigenvalues $\lambda_1 = 0.5$ and $\lambda_2=0.1$. What happens to $A^k$ as $k$ becomes very large?