  Given a symmetric matrix $M$ and a vector $\lambda$, we present new bounds on
the Frobenius-distance utility of the Gaussian mechanism for approximating $M$
by a matrix whose spectrum is $\lambda$, under
$(\varepsilon,\delta)$-differential privacy. Our bounds depend on both
$\lambda$ and the gaps in the eigenvalues of $M$, and hold whenever the top
$k+1$ eigenvalues of $M$ have sufficiently large gaps. When applied to the
problems of private rank-$k$ covariance matrix approximation and subspace
recovery, our bounds yield improvements over previous bounds. Our bounds are
obtained by viewing the addition of Gaussian noise as a continuous-time matrix
Brownian motion. This viewpoint allows us to track the evolution of eigenvalues
and eigenvectors of the matrix, which are governed by stochastic differential
equations discovered by Dyson. These equations allow us to bound the utility as
the square-root of a sum-of-squares of perturbations to the eigenvectors, as
opposed to a sum of perturbation bounds obtained via Davis-Kahan-type theorems.
