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

For a nonzero positive semidefinite covariance matrix $\Sigma$, its covariance effective rank iswhere the operator norm is induced by Euclidean norm and the ${\lambda }_j$ are the nonnegative eigenvalues. Each nonzero eigenvalue contributes at most one to this ratio, soA scalar multiple of $I_p$ has covariance effective rank $p$, while a matrix $\lambda u{u}^T$ with $\lambda >0$ and a unit vector $u$ has covariance effective rank one. For $\Sigma =\mathrm{diag}(100,1,1)$, the ratio is $102/100=1.02$, although the ordinary matrix rank is three.

Thus the quantity measures total variance relative to the largest directional variance, rather than merely counting nonzero directions. Multiplying $\Sigma$ by a positive scalar leaves the ratio unchanged. Its role in the Gaussian sample-covariance operator-norm bound makes it a useful measure of the dimension that matters for covariance estimation. The zero matrix requires a separate convention because the displayed ratio is $0/0$.

Let $X_1,\dots ,X_n\in {\mathbb{R}}^p$ be independent centered Gaussian random vectors with common nonzero covariance matrix $\Sigma$, and put $\hat{\Sigma }=n^{-1}{\sum }_i{X}_i{X}_i^T$. There is a universal constant $C>0$ such that for $t\ge 1$, with probability at least $1-e^{-t}$,where $r(\Sigma )=\mathrm{tr}(\Sigma )/\Vert \Sigma {\Vert }_{\mathrm{op}}$ is the effective rank of a covariance matrix.

The bound combines an expected operator norm error with Gaussian concentration inequality. In Euclidean space, the Gaussian effective-dimension parameter in the primary result is comparable to this trace ratio; replacing maxima by sums gives the displayed version.^[ref]

For $\Sigma =I_p$, the covariance effective rank is $p$, and relative error is controlled by $\sqrt{(p+t)/n}+(p+t)/n$. covariance concentrated in fewer directions can have much smaller covariance effective rank. The estimator here is the uncentered empirical second moment for a known zero mean; an unknown mean and empirical centering require the corresponding formulation. The bound remains meaningful when $\Sigma$ is singular but nonzero, since it does not invert $\Sigma$.