Exercise 8

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

1. Show that the eigenvalues of $A$ are complex.

2. Choose one of the complex eigenvalues $\lambda=a+bi$ and construct the usual matrix $C$.

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 $\vvec$. Write $\vvec = \vvec_1 - i \vvec_2$ to identify vectors $\vvec_1$ and $\vvec_2$ having real entries.

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