# Special Eigenvalues

• Nov 3rd 2012, 08:15 PM
sfspitfire23
Special Eigenvalues
Hi all, I have been asked to diagonalize the matrix [-1,3 ; 3,2] and I am supposed to notice something special about the eigenvectors.

Eigenvectors are as follows:

[-1.61803 ; 1] and [.61803 ; 1]. They are independent etc....not sure what is so special about them...

Thanks.
• Nov 3rd 2012, 09:37 PM
chiro
Re: Special Eigenvalues
Hey sfspitfire23.

I'm not exactly sure about the context of your question, but those vectors are independent.

Don't know if that helps or not.
• Nov 4th 2012, 12:45 AM
Deveno
Re: Special Eigenvalues
it might help if you wrote the ACTUAL values for the eigenvectors, and not just decimal approximations (decimals are BAD! unless you're a physicist. silly them).

i get as eigenvalues:

$\lambda_1 = \frac{1+3\sqrt{5}}{2},\ \lambda_2 = \frac{1-3\sqrt{5}}{2}$

leading to the following eigenvectors:

$v_1 = \begin{bmatrix}1\\ \frac{1+\sqrt{5}}{2} \end{bmatrix},\ v_2 = \begin{bmatrix}1\\ \frac{1-\sqrt{5}}{2} \end{bmatrix}$.

note that:

$v_1 \cdot v_2 = 1^2 + \left(\frac{1+\sqrt{5}}{2}\right)\left(\frac{1-\sqrt{5}}{2}\right) = 1 + \frac{1 - 5}{4} = 0$

in other words, the eigenvectors are ORTHOGONAL.
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