# Complex eigenvalues

### Complex eigenvalues

#### Lessons

In this section, we will be finding complex eigenvalues and eigenvectors of $2 \times 2$ matrices. We will not go any higher than a $2 \times 2$.

Given a complex vector , the real part and imaginary part would be:

Let $x$$Ax$ be a transformation that is composed of scaling and rotation. Also, let . Then we can find the angle (argument) $\varphi$ in the complex plane by doing the following steps:

1. Find the complex eigenvalues
2. Find the scale factor $| \lambda |=r=\sqrt{a^2+b^2}$
3. Divide the scale factor of the matrix
4. Compare with the rotational matrix and create two equations
5. Solve for the argument $\varphi$

Let $A$ be a real $2 \times 2$ matrix with a complex eigenvalue $\lambda =a-bi$ and associated eigenvector $v$ in $\Bbb{C}^2$. Then $A=PCP^{-1}$ where

• 1.
Complex Eigenvalues Overview:
a)
Review of Complex numbers
• The square root of -1
• Finding complex roots using the quadratic equation
• Re$(v)$ and Im$(v)$ of Complex vectors

b)
Complex Eigenvalues and Eigenvectors
• The Characteristic Equation
• Using the quadratic formula to find the complex roots
• The corresponding eigenvectors

c)
Finding the Argument
• Find the Eigenvalues $a \pm bi$
• Find the scale factor $| \lambda |=r=\sqrt{a^2+b^2}$
• Divide the scale factor of the matrix
• Compare with the rotational matrix
• Solve for the argument $\varphi$

d)
The Formula $A=PCP^{-1}$
• Get the Eigenvalue $\lambda =a-bi$ and Eigenvector $v$
• Find Re$(v)$ and Im$(v)$
• Combine them to get the invertible matrix $P$
• Use $a$ and $b$ from the eigenvalue $a-bi$ to get matrix $C$

• 2.
Finding the Complex Eigenvalues/Eigenvectors
Find the complex eigenvalues of A and their corresponding eigenvectors.

• 3.
Finding the Argument using Eigenvalues
The transformation $x$$Ax$ is the composition of a rotation and scaling. Find the eigenvalues of $A$. Then give the angle $\varphi$ of the rotation, where $-\pi$ < $\varphi \leq \pi$, and give the scale factor $r$ if

• 4.
The transformation $x$$Ax$ is the composition of a rotation and scaling. Find the eigenvalues of $A$. Then give the angle $\varphi$ of the rotation, where $-\pi$ < $\varphi \leq \pi$, and give the scale factor $r$ if

• 5.
Finding the Invertible Matrix and Matrix C
Find an invertible matrix $P$ and a matrix $C$ of the form such that the given matrix has the form $A=PCP^{-1}$ if

• 6.
Advanced Proofs Related to the Eigenvector
Let $A$ be a real $n \times n$ matrix, and let $x$ be a vector in $\Bbb{C}^n$. Show that

Re$(Ax)=A($Re $x)$

Im$(Ax)=A($Im $x)$

• 7.
Let $A$ be a real $2 \times 2$ matrix with a complex eigenvalue $\lambda =a-bi$ where $b \neq 0$ and an associated eigenvector $v$ in $\Bbb{C}^2$. Show that:
$A($Re $v)=aRe\; v+b$Im $v$
$A($Im $v)=-bRe\; v+a$Im $v$