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

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.