Activity 4.3.4
  1. 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 \(k\text{?}\)

  2. 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 \(\lambda^2\text{.}\)

  3. 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.

  4. 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*}
  5. In the same way, find a diagonalization of \(A^3\text{,}\) \(A^4\text{,}\) and \(A^k\text{.}\)

  6. 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?