complex_eigenvalues.tex

\section{Complex Eigenvalues}
\begin{theorem}
\begin{enumerate} Complex Eigenvalues
    \item If \(\lambda \in \mathbb{C}\) is a root of a real polynomial \(p(x)\), then the conjugate \(\lambda\) is also a root of \(p(x)\).
    \item If \(\lambda\) is an eigenvalue of real matrix \(A\) with eigenvector \(\Vec{v}\), then \(\Bar{\lambda}\) is an eigenvalue of \(A\) with eigenvector \(\Vec{v}\).
\end{enumerate}
\end{theorem}

\section{Rotation-Dilation Matrix}
\begin{align}
    A &= \begin{pmatrix}
        a & -b \\ b & a
    \end{pmatrix} \\
    &= \begin{pmatrix}
        r & 0 \\ 0 & r
    \end{pmatrix}
    \begin{pmatrix}
        cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)
    \end{pmatrix}
\end{align}

\noindent
The eigenvalues of \(A\) are \(a \pm bi\).