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 $\forall \beta \ge n-2$ 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 $A_n:=X_n{X}_n^T$, where $X_n$ is $n\times n$ matrix with i.i.d. Gaussians. It is shown that applying $x\to |x{|}^{\alpha }$ 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 $\alpha =s$.

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