Activity 4.3.5
  1. We will rewrite \(C\) in terms of \(r\) and \(\theta\text{.}\) Explain why

    \begin{equation*} \left[\begin{array}{rr} a \amp -b \\ b \amp a \\ \end{array}\right] = \left[\begin{array}{rr} r\cos\theta \amp -r\sin\theta \\ r\sin\theta \amp r\cos\theta \\ \end{array}\right] = \left[\begin{array}{rr} r \amp 0 \\ 0 \amp r \\ \end{array}\right] \left[\begin{array}{rr} \cos\theta \amp -\sin\theta \\ \sin\theta \amp \cos\theta \\ \end{array}\right]\text{.} \end{equation*}
  2. Explain why \(C\) has the geometric effect of rotating vectors by \(\theta\) and stretching them by a factor of \(r\text{.}\)

  3. Let's now consider the matrix \(A\) from Example 4.3.8:

    \begin{equation*} A = \left[\begin{array}{rr} -2 \amp 2 \\ -5 \amp 4 \\ \end{array}\right] \end{equation*}

    whose eigenvalues are \(\lambda_1 = 1+i\) and \(\lambda_2 = 1-i\text{.}\) We will choose to focus on one of the eigenvalues \(\lambda_1 = a+bi= 1+i. \)

    Form the matrix \(C\) using these values of \(a\) and \(b\text{.}\) Then rewrite the point \((a,b)\) in polar coordinates by identifying the values of \(r\) and \(\theta\text{.}\) Explain the geometric effect of multiplying vectors of \(C\text{.}\)

  4. Suppose that \(P=\left[\begin{array}{rr} 1 \amp 1 \\ 2 \amp 1 \\ \end{array}\right] \text{.}\) Verify that \(A = PCP^{-1}\text{.}\)

  5. Explain why \(A^k = PC^kP^{-1}\text{.}\)

  6. We formed the matrix \(C\) by choosing the eigenvalue \(\lambda_1=1+i\text{.}\) Suppose we had instead chosen \(\lambda_2 = 1-i\text{.}\) Form the matrix \(C'\) and use polar coordinates to describe the geometric effect of \(C\text{.}\)

  7. Using the matrix \(P' = \left[\begin{array}{rr} 1 \amp -1 \\ 2 \amp -1 \\ \end{array}\right] \text{,}\) show that \(A = P'C'P'^{-1}\text{.}\)