Spectral decomposition of Covariance matrix

Question

Let, $\Sigma$ = $ \begin{pmatrix} \sigma^{2} & \sigma^{2}\rho \\ \sigma^{2}\rho & \sigma^{2} \end{pmatrix}$

What i want to find is spectral decomposition of $\Sigma$.

What I can't do is getting eigenvalue and eigenvector. Using $\lvert A-\lambda I \rvert$, I stuck simple equation $\sigma^{4}(1-\rho)^{2} =\lambda(\lambda-2\sigma^{2})$ . How can I find values of $\lambda$.

Thanks advance you guys.

Answer 1

One has $0 = det(\Sigma-\lambda I) = (\sigma^2-\lambda)^2 - \sigma^4\rho^2 = (\sigma^2-\sigma^2\rho - \lambda)(\sigma^2+\sigma^2\rho - \lambda)$.

Then, the eigenvalues are $\lambda_1 = \sigma^2\rho -\sigma^2$, $\lambda_2 = -\sigma^2\rho -\sigma^2$.