Ch05 Eigenvalues and Eigenvectors

5.5 Complex Eigenvalues

Complex Eigenvalues

• The matrix eigenvalue-eigenvector theory already developed for $\mathcal{R}^n$ applies equally well $\mathcal{C}^n$.
• So a complex scalar $\lambda$ satisfied $\text{det }(A-\lambda I)=0$ if and only if there is a nonzero vector $\textbf{x} \in \mathbb{C}^n$ such that $A \textbf{x} = \lambda \textbf{x}$.
• We call $\lambda$ a (complex) eigenvalue and $\textbf{x}$ a (complex) eigenvector corresponding to $\lambda$.

Example 1:

If $A=\begin{bmatrix}0 & -1 \\ 1 & 0 \end{bmatrix}$, then the linear transformation $\textbf{x} \mapsto A\textbf{x}$ on $\mathbb{R}^2$ rotates the plane counterclockwise through a quater-turn.

• The action of $A$ is periodic, since after four quarter-turns, a vector is back where it started.
• Obviously, no nonzero vector is mapped into a multiple of itself, so $A$ has no eigenvectors in $\mathbb{R}^2$ and hence no real eigenvalues.
• In fact, the characteristic equation of $A$ is $$\lambda^2+1=0$$
• The only roots are complex: $\lambda= i \text{ and } \lambda = -i$. However, if we permit $A$ to act on $\mathcal{C}^2$, then $$\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ -i \end{bmatrix} = \begin{bmatrix} i \\ 1 \end{bmatrix} = i \begin{bmatrix} 1 \\ -i \end{bmatrix} \\ \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ i \end{bmatrix} = \begin{bmatrix} i \\ 1 \end{bmatrix} = i \begin{bmatrix} 1 \\ -i \end{bmatrix}$$
• Thus $i$ and $-i$ are eigenvalues, with $\begin{bmatrix}1 \\ -i \end{bmatrix}$ and $\begin{bmatrix}1 \\ i \end{bmatrix}$ as corresponding eigenvectors. $\blacksquare$

Real and Imaginary Parts of Vectors

• The complex conjugate of a complex vector $\textbf{x} \in \mathbb{C}^n$ is the vector $\bar{\textbf{x}} \in \mathbb{C}^n$ whose entries are the complex conjugate of the entries in $\textbf{x}$.
• The real and imaginary parts of a complex vector $\textbf{x}$ are the vectors $\text{Re }\textbf{x}$ and $\text{Im }\textbf{x}$ in $\mathbb{R}^n$ formed from the real and imaginary parts of the entries of $\textbf{x}$.

Example 4:

If $\textbf{x}=\begin{bmatrix} 3-i \\ i \\ 2+5i \end{bmatrix} = \begin{bmatrix} 3 \\ 0 \\ 2 \end{bmatrix} + i \begin{bmatrix} -1 \\ 1 \\ 5 \end{bmatrix}$, then

$$\text{Re }\textbf{x}=\begin{bmatrix} 3 \\ 0 \\ 2 \end{bmatrix}, \text{Im }\textbf{x}=\begin{bmatrix} -1 \\ 1 \\ 5 \end{bmatrix}, \text{ and } \bar{\textbf{x}} = \begin{bmatrix} 3 \\ 0 \\ 2 \end{bmatrix} -i \begin{bmatrix} -1 \\ 1 \\ 5 \end{bmatrix} = \begin{bmatrix} 3+i \\ -i \\ 2-5i \end{bmatrix}$$

Theorem 9:

Let $A$ be a real $2 \times 2$ matrix with a complex eigenvalue $\lambda=a-bi (b\ne 0)$ and an associated eigenvector $\textbf{v}$ in $\mathbb{C}^2$. Then $$A= PCP^{-1}, \\ $$ ,where $P=\begin{bmatrix} \text{Re }\textbf{v} & \text{Im } \textbf{v} \end{bmatrix}$ and $C=\begin{bmatrix} a & -b \\ b & a \end{bmatrix}$