# Eigenvector with complex eigenvalues

• Nov 7th 2010, 08:01 PM
acevipa
Eigenvector with complex eigenvalues
I have to find the eigenvalues and eigenvectors for this matrix:

$\displaystyle \begin{pmatrix}4&2i\\2i&6\end{pmatrix}$

I found eigenvalues of $\displaystyle \lambda=5\pm\sqrt{3}i$

I'm having trouble finding the eigenvectors

$\displaystyle ker\begin{pmatrix}-1\pm\sqrt{3}i&2i\\2i&1\pm\sqrt{3}i\end{pmatrix}$

I know how to find eigenvectors, but this one seems confusing because of the complex eigenvalues
• Nov 7th 2010, 08:35 PM
Drexel28
Quote:

Originally Posted by acevipa
I have to find the eigenvalues and eigenvectors for this matrix:

$\displaystyle \begin{pmatrix}4&2i\\2i&6\end{pmatrix}$

I found eigenvalues of $\displaystyle \lambda=5\pm\sqrt{3}i$

I'm having trouble finding the eigenvectors

$\displaystyle ker\begin{pmatrix}-1\pm\sqrt{3}i&2i\\2i&1\pm\sqrt{3}i\end{pmatrix}$

I know how to find eigenvectors, but this one seems confusing because of the complex eigenvalues

Just do what's natural suppose that $\displaystyle \begin{bmatrix}z_1\\ z_2\end{bmatrix}\in\mathbb{C}^2$ is an eigenvector, then: $\displaystyle \begin{bmatrix}4 & 2i\\ 2i &6\end{bmatrix}\begin{bmatrix}z_1\\ z_2\end{bmatrix}=\begin{bmatrix}(5+\sqrt{3}i)z_1\\ (5+\sqrt{3}i)z_2\end{bmatrix}$. Expand this and solve.