# Find Eigenvectors in a 3x3-matrix

• Sep 18th 2013, 10:32 AM
aprilrocks92
Find Eigenvectors in a 3x3-matrix
Hi,

I'm having trouble with finding the eigenvectors of a 3x3 matrix. The matrix M, is defined as the following:
 0 0.3 0.4 0.6 0 0.6 0.4 0.7 0

Having found det(M - ÆI), where ÆI is matrix:
 Æ 0 0 0 Æ 0 0 0 Æ

I have found the following eigenvalues:
Æ1: 1
Æ2: -0.6
Æ3: -0.4

I am now to find the corresponding eigenvectors, V1, V2, V3, but I can't seem to approach it the right way. I have the following system of equations:

For Æ1 = 1:
 0.3y + 0.4z = x 0.6x + 0.6z = y 0.4x + 0.7y = z

And similarly for the remaining eigenvalues.

How do I go about finding the eigenvectors? ANY help is highly appreciated!
• Sep 18th 2013, 07:21 PM
DavidB
Re: Find Eigenvectors in a 3x3-matrix
Given the eigenvalues of a matrix, (by definition) to find the corresponding eigenvectors, solve the following system:

$\begin{bmatrix} a11-\lambda & a12& a13 \\ a21& a22-\lambda & a23\\ a31& a32& a33-\lambda \end{bmatrix}\begin{bmatrix}x1 \\ x2\\ x3\end{bmatrix} = 0$

So, for the first eigenvalue, λ = 1, the system would become as follows:

$\begin{bmatrix} -1& 0.3& 0.4 \\ 0.6& -1& 0.6\\ 0.4& 0.7& -1 \end{bmatrix}\begin{bmatrix}x1 \\ x2\\ x3\end{bmatrix} = 0$

Solve this equation.

Then repeat this process for the other two eigenvalues to find the corresponding eigenvectors.