Real Matrixes with Complex Eigenvalues
Given a $2 \times 2$ matrix in the form:
$$ A = \begin{bmatrix} a & -b \\b & a \end{bmatrix} $$
This is a rotational transformation
Then $A$ has the eigenvalues $\lambda_{\pm}= a \pm bi$ and can be represented by the Euler’s Identity of
$$ a + bi = re^{i\phi} $$
$$ A = r\begin{bmatrix} cos \phi & -sin \phi \\sin \phi & cos \phi \end{bmatrix} $$
Diagonalizing the matrix $B$, which has the eigenvalues : $a \pm bi$ is calculated using the method below:
$$ B = P\begin{bmatrix} a & -b \\b & a \end{bmatrix} P^{-1} $$
$$ P = \begin{bmatrix} Re(\mathbf{v}) & Im(\mathbf{v}) \end{bmatrix} $$
where $\mathbb{v}$ is the eigenvector for $a - bi$