Inference about heterogeneity of sample from normal population looking at covariance matrix

Question

consider:

$Y_1,\dots,Y_n \sim N(0,\sigma^2)$ iid.

If I have a high variance I will see a very heterogenous sample and a large bell around the mean. It is sufficient to look at $\sigma^2$ to have an idea about it.

Now consider:

$\mathbf{Y_1},\dots,\mathbf{Y_n} \sim N_p(\mathbf{0},\Sigma)$.

My question is simple:

How do I infer the 'heterogeneity' of samples by looking at the covariance matrix?

Is it possible to say something in general by looking for example at the determinant of the matrix (not considering the $p=2$ case)? Or should I just look at the single entries of the matrix?

Thank you in advance

Answer 1

Heterogeneity is large variance? If so, then for $Y$ that is MVN with $\Sigma$ as a covariance matrix, its main diagonal is the variance of each component of $Y$, where the non-diagonal terms are covariance of $y_i$ and $y_j$, as such they can be very "large" due to high variance or/and strong correlation. Besides, any judgment as large or small depends on your application. Looking at its determinant can be useful for some purposes as $|\Sigma| = \prod_{i=1}^n\lambda_i$ when the eigenvalues can give you some indication of the dominant elements.