# Complex eigenvalues

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##### Intros

###### Lessons

**Complex Eigenvalues Overview:**__Review of Complex numbers__

• The square root of -1

• Finding complex roots using the quadratic equation

• Re$(v)$ and Im$(v)$ of Complex vectors__Complex Eigenvalues and Eigenvectors__

• The Characteristic Equation

• Using the quadratic formula to find the complex roots

• The corresponding eigenvectors__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 $\varphi$__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

**Finding the Complex Eigenvalues/Eigenvectors**

Find the complex eigenvalues of A and their corresponding eigenvectors.

**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

- 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

**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

**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 $x)$ - 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 $v$