# Eigenvalues of time lagged covariance matrix - should be always real and positive?

Consider two random variables $$X_t$$ and $$X_{t+\tau}$$; where both come from random variable $$X$$ but are lagged by $$\tau$$. Assume that both are mean-free. If we want calculate covariance matrix of them, then we do:

$$\Sigma = X_t^T X_{t+\tau}$$ if we perform eigendecomposition

$$A\Sigma = \Sigma\Lambda$$

Then if the time lagged covariance is properly converged, all the eigenvalues are going to be positive, non-complex ?

• What do you mean by "properly converged"? Nov 25 at 19:07
• @mhdadk There's enough samples to calculate it, so that it does not change much if we add more samples Nov 25 at 20:12
• One could check for diagonal dominance, see math.stackexchange.com/questions/87528/…. By intuition, if autocorrelation $<X_{t}, X_{t+\tau}>$ is much higher than the rest, it is probable that $\Sigma$ is positive definite. Nov 25 at 23:36