Assume that the following is used:
$$ A = \begin{pmatrix} 0& 1&\\ 2& 3&\\ 4& 5&\\ 6& 7&\\ 8& 9& \end{pmatrix} $$
Then calculating the Coveriance matrix, which, gives me:
$$ C = \begin{pmatrix} 40& 40&\\ 40& 40&\\ \end{pmatrix} $$
Then using the following:
$$ det = (a+b) \cdot (a+b)-4 \cdot(a \cdot b - c \cdot c), $$
where in this case, $a = 40, b = 40, c = 40$ gives the answer:
$$ \lambda_{1} = 80, \\ \lambda_{2} = 0, $$
These are therefore the correct Eigen values. However, using this formula, if I have the following:
$$ A = \begin{pmatrix} -4& -2&\\ -1& -3&\\ 4& 5&\\ 6& 7&\\ 8& 9& \end{pmatrix}, $$
where the Covariance matrix is given:
$$ C = \begin{pmatrix} 99.2& 103.4&\\ 103.4& 116.8&\\ \end{pmatrix}, $$
gives the Eigenvalues as:
$$ \lambda_{1} = 218.119 \\ \lambda_{2} = -15.5189 $$
When the actual values are: $$ \lambda_{1} =211.774 \\ \lambda_{2} = 4.226 $$
Could anyone tell me where I am calculating this wrong please?
EDIT:
For $\lambda_{1} = (a + b + det)/2 \\ \lambda_{2} = (a + b - det)/2 $
Have you double-checked your calculations? The eigenvalues of any $2\times2$ real symmetric matrix $\pmatrix{a&c\\ c&b}$ are given by $$\frac{a+b\pm\sqrt{(a+b)^2-4(ab-c^2)}}2.$$ Plug in the entries of $C$, I don't find any discrepancies.