I'm following a paper found here: (https://bearspace.baylor.edu/Vince_Cronin/www/PBO_ed/FindingEigenvectors.pdf)
It states that if you have two given eigen values $\lambda_{1}$ and $\lambda_{2}$ then you can calculate the the eigenvectors by: $$ \frac{1}{ \sqrt{1^{2} + (\frac{\lambda_{1} - \alpha}{3}})^{2}} $$
Where $\lambda_{1}$ is the largest eigenvalue. I've tried with the example, and, this makes sense, but using the following covariance matrix:
$$ C = \begin{pmatrix} 4.5 & 4.5 \\ 4.5 & 4.5 \end{pmatrix}, $$
where the Eignvalues are given as $\lambda_{1} = 9.0, \lambda_{2} = 0.0$
I do the following:
$$ \alpha = ((9.0 - 4.5)/3 \cdot (9.0 - 4.5)/3 + (1 \cdot 1), $$
and then:
$$ \beta = (1 / sqrt(\alpha)) $$
The result I get is: $\lambda_{1} = 0.5547$ when in actual fact the eigenvectors for this matrix is given as: $\lambda_{1} = 0.707$
I can't understand where I'm going wrong and I am probably missing something stupid
The $3$ is from his worked example matrix.
For your problem, you should change $3$ to $4.5$.
You get:
$$ \alpha = ((9.0 - 4.5)/34.5\cdot (9.0 - 4.5)/4.5 + (1 \cdot 1) = 2, $$
and then:
$$ \beta = \dfrac{1}{\sqrt\alpha} = \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$$