# Complex eigenvalues

##### Intros
###### Lessons
1. Complex Eigenvalues Overview:
2. Review of Complex numbers
• The square root of -1
• Finding complex roots using the quadratic equation
• Re$(v)$ and Im$(v)$ of Complex vectors
3. Complex Eigenvalues and Eigenvectors
• The Characteristic Equation
• Using the quadratic formula to find the complex roots
• The corresponding eigenvectors
4. 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 $\varphi$
5. 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 $C$
##### Examples
###### Lessons
1. Finding the Complex Eigenvalues/Eigenvectors
Find the complex eigenvalues of A and their corresponding eigenvectors.
1. Finding the Argument using Eigenvalues
The transformation $x$$Ax$ is the composition of a rotation and scaling. Find the eigenvalues of $A$. Then give the angle $\varphi$ of the rotation, where $-\pi$ < $\varphi \leq \pi$, and give the scale factor $r$ if
1. The transformation $x$$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
1. 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
1. 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 $x)$
1. 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 $v$