When \(A\) is a \(2\times2\) matrix with a complex eigenvalue \(\lambda = a+bi\text{,}\) we have said that there is a matrix \(P\) such that \(A=PCP^{-1}\) where \(C=\left[\begin{array}{rr} a \amp -b \\ b \amp a \\ \end{array}\right] \text{.}\) In this exercise, we will learn how to find the matrix \(P\text{.}\) As an example, we will consider the matrix \(A = \left[\begin{array}{rr} 2 \amp 2 \\ -1 \amp 4 \\ \end{array}\right] \text{.}\)

  1. Show that the eigenvalues of \(A\) are complex.

  2. Choose one of the complex eigenvalues \(\lambda=a+bi\) and construct the usual matrix \(C\text{.}\)

  3. Using the same eigenvalue, we will find an eigenvector \(\vvec\) where the entries of \(\vvec\) are complex numbers. As always, we will describe \(\nul(A-\lambda I)\) by constructing the matrix \(A-\lambda I\) and finding its reduced row echelon form. In doing so, we will necessarily need to use complex arithmetic.

  4. We have now found a complex eigenvector \(\vvec\text{.}\) Write \(\vvec = \vvec_1 - i \vvec_2\) to identify vectors \(\vvec_1\) and \(\vvec_2\) having real entries.

  5. Construct the matrix \(P = \left[\begin{array}{rr} \vvec_1 \amp \vvec_2 \end{array}\right]\) and verify that \(A=PCP^{-1}\text{.}\)