On Hadamard powers of random Wishart matrices

Abstract

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