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:

I have found the following **eigenvalues**:

**Æ**_{1}: 1

**Æ**_{2}: -0.6

**Æ**_{3}: -0.4

I am now to find the corresponding **eigenvectors, V**_{1}, V_{2}, V_{3}, but I can't seem to approach it the right way. I have the following system of equations:

For **Æ**_{1} = 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!

Re: Find Eigenvectors in a 3x3-matrix

Given the eigenvalues of a matrix, (by definition) to find the corresponding eigenvectors, solve the following system:

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

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

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

Solve this equation.

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