# Compute the limit of trace of inverse of square of rank-1 perturbation of Wishart matrix

Let $$a \ge 0$$, $$b,c>0$$ be fixed constants, and let $$X$$ be an $$m \times d$$ random matrix with entries drawn iid from $$N(0,1/d)$$. Consider the random psd matrix $$S := a 1_m 1_m^\top + b XX^\top + c I_m$$.

Question 1. In the limit $$m,d \to \infty$$ with $$m/d \to \rho \in (0,\infty)$$, what does $$d^{-1}\mbox{trace}(S^{-2})$$ converge to ?

Observation. If $$a = 0$$, then one computes $$d^{-1}\mbox{trace}(S^{-2}) = \frac{1}{d}\sum_{i=1}^n \frac{1}{(bXX^\top +c)^2} \overset{a.s}{\to} b^{-2}m'_{MP(\rho)}(-c/b),$$ where $$m_{MP(\rho)}$$ is the Stieltjes transform of the Marchenko-Pastur distribution with parameter $$\rho$$.

## Update

I've often heard that

"Finite-rank perturbations don't change limiting empirical spectral distribution of random matrices."

Unfortunately, I can't find a definitive reference for this statement.

Question 2. In view of the previous remark, is it true that $$d^{-1}\mbox{trace}(S^{-2}) \overset{a.s}{\to} b^{-2}m'_{MP(\rho)}(-c/b)$$ for all $$a \ge 0$$ and $$b,c>0$$ ?

Let me try to answer the updated question. If $$A$$ and $$B$$ are Hermitian $$m\times m$$ matrices with $$B$$ of rank $$r$$ then the two sequences of eigenvalues $$\lambda_k(A+B)$$ of $$A+B$$ and $$\lambda_k(A)$$ of $$A$$, each sorted in ascending order, are related by $$\lambda_k(A)\leq\lambda_{k+r}(A+B)\leq\lambda_{k+2r}(A),\;\;1\leq k\leq m-2r.$$ For a proof see Theorem 8 of these notes. In this case $$A=S(a)$$ differs from $$A+B=S(0)$$ by a rank-one matrix, $$r=1$$, so the eigenvalues of $$S(a)$$ are interlaced with those of $$S(0)$$. Their spectral densities in the large-$$m$$ limit must coincide and Question 2 can be answered in the affirmative.