Activity 4.3.4

Let's begin with the diagonal matrix
\begin{equation*} D = \left[\begin{array}{rr} 2 \amp 0 \\ 0 \amp 1 \\ \end{array}\right]\text{.} \end{equation*}Find the powers \(D^2\text{,}\) \(D^3\text{,}\) and \(D^4\text{.}\) What is \(D^k\) for a general value of \(k\text{?}\)
Suppose that \(A\) is a matrix with eigenvector \(\vvec\) and associated eigenvalue \(\lambda\text{;}\) that is, \(A\vvec = \lambda\vvec\text{.}\) By considering \(A^2\vvec\text{,}\) explain why \(\vvec\) is also an eigenvector of \(A\) with eigenvalue \(\lambda^2\text{.}\)

Suppose that \(A= PDP^{1}\) where
\begin{equation*} D = \left[\begin{array}{rr} 2 \amp 0 \\ 0 \amp 1 \\ \end{array}\right]\text{.} \end{equation*}Remembering that the columns of \(P\) are eigenvectors of \(A\text{,}\) explain why \(A^2\) is diagonalizable and find a diagonalization of it.

Give another explanation of the diagonalizability of \(A^2\) by writing
\begin{equation*} A^2 = (PDP^{1})(PDP^{1}) = PD(P^{1}P)DP^{1}\text{.} \end{equation*} In the same way, find a diagonalization of \(A^3\text{,}\) \(A^4\text{,}\) and \(A^k\text{.}\)
Suppose that \(A\) is a diagonalizable \(2\times2\) matrix with eigenvalues \(\lambda_1 = 0.5\) and \(\lambda_2=0.1\text{.}\) What happens to \(A^k\) as \(k\) becomes very large?