Exercise8

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 $$A=PCP^{-1}\text{.}$$

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