On Hadamard powers of Random Wishart matrices

Abstract: A famous result of Horn and Fitzgerald is that the $\beta$-th Hadamard power of any $n\times n$ positive semi-definite (p.s.d) matrix with non-negative entries is p.s.d $\forall \beta\geq n-2$ and is not necessarliy p.s.d for $\beta< n-2,$ with $\ \beta\notin \mathbb{N}$. In this article, we study this question for random Wishart matrix $A_n:={X_nX_n^T}$, where $X_n$ is $n\times n$ matrix with i.i.d. Gaussians. It is shown that applying $x\rightarrow |x|^{\alpha}$ entrywise to $A_n$, the resulting matrix is p.s.d, with high probability, for $\alpha>1$ and is not p.s.d, with high probability, for $\alpha<1$. It is also shown that if $X_n$ are $\lfloor n^{s}\rfloor\times n$ matrices, for any $s<1$, the transition of positivity occurs at the exponent $\alpha=s$.