Given a symmetric matrix M and a vector λ, we present new bounds on the Frobenius-distance utility of the Gaussian mechanism for approximating M by a matrix whose spectrum is λ, under (ε,δ)-differential privacy. Our bounds depend on both λ 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…