Effective rank of a covariance matrix ($r(\Sigma )=\mathrm{tr}(\Sigma )/\Vert \Sigma {\Vert }_{\mathrm{op}}$)

The effective rank measures the total variance of a covariance matrix relative to its largest directional variance.

For independent centered Gaussian vectors with covariance $\Sigma$ and uncentered sample covariance $\hat{\Sigma }=n^{-1}{\sum }_i{x}_i{x}_i^T$, with probability at least $1-e^{-t}$,