  We prove a central limit theorem for the difference of linear eigenvalue
statistics of a sample covariance matrix $\widetilde{W}$ and its minor $W$. We
find that the fluctuation of this difference is much smaller than those of the
individual linear statistics, as a consequence of the strong correlation
between the eigenvalues of $\widetilde{W}$ and $W$. Our result identifies the
fluctuation of the spatial derivative of the approximate Gaussian field in the
recent paper by Dumitru and Paquette. Unlike in a similar result for Wigner
matrices, for sample covariance matrices the fluctuation may entirely vanish.
