I'm working on some Random Matrix Theory related stuff for my thesis, and i've come across the following problem:

Consider a (normalized) spiked Wigner matrix $\mathbf{A}$
$$ \mathbf{A} = \frac{\beta}{n}\mathbf{xx}^T + \mathbf{W}, $$ 
where $\mathbf{x}$ is an n-dimensional vector (in our case it's on the Boolean hypercube) such that $\|x\|_2 = \sqrt{n}$, $\beta < 1$, and 
$$ \mathbf{W}_{ij} \sim 
\begin{cases}
\mathcal{N}(0,1/n), \quad i\neq j \\
\mathcal{N}(0,2/n), \quad i = j.
\end{cases}
$$

I'm trying to show that it is asymptotically unitarily invariant (or not, perhaps). My intuition is that by the [BBP phase transition][1], the spectrum of $\mathbf{A}$ remains unfazed by the spike (i.e. it's the [Wigner semicircle distribution][2]). 

By unitary invariance, I mean that I want to show that for a Haar distributed matrix $\mathbf{V}$, we can write out 
$$ \mathbf{A} = \mathbf{V\Lambda V}^T $$
in the large system limit $n \rightarrow \infty$, where $\boldsymbol{\Lambda}$ is the diagonal containing the eigenvalues of $\mathbf{W}$.

Any ideas on how to show this property?

Thanks



  [1]: https://arxiv.org/pdf/math/0403022
  [2]: https://en.wikipedia.org/wiki/Wigner_semicircle_distribution