Abstract

The celebrated Sherman–Morrison rank-one-perturbation formula and its rank-k-perturbation extension are useful in statistics, linear algebra, optimization, and theoretical physics. We provide an extension of the symmetric Sherman–Morrison formula for an inverse of $A+vv’$ to an inverse of multiple rank-one-perturbations $A+v_1v_1’+v_2v_2’+…+ v_pv_p’$ for $p>1$. The result is proved by induction in $p$. We also work out formulas for bilinear forms for pairs $(v_i, v_j)$ with respect to such an inverse.

We provide an application of these formulas to derivation of asymptotic behavior of off-diagonal elements of projection matrices, whose underlying dimension $m$ is asymptotically proportional to sample size $n$, so that $m/n\to\mu$ as $n\to\infty$, where the aspect ratio $\mu\in (0,1)$. Under independence design and using the random matrix theory, we derive the joint asymptotic distribution for a collection of their off-diagonal elements, whose indices, one or both, may or may not coincide. The rate of convergence turns out to be $\sqrt n$, and the limiting distribution turns out to be multivariate centered Gaussian with an interesting pattern in the asymptotic variance-covariance matrix.