On fractional Hadamard powers of positive block matrices

Abstract: Entrywise powers of matrices have been well-studied in the literature, and have recently received renewed attention in the regularization of high-dimensional correlation matrices. In this paper, we study powers of positive semidefinite block matrices $(H_{st}){s,t=1}^n$ with complex entries. We first characterize the powers $\alpha\in\mathbb{R}$ such that the blockwise power map $(H{st}) \mapsto (H_{st}^\alpha)$ preserves Loewner positivity. The characterization is obtained by exploiting connections with the theory of matrix monotone functions developed by Loewner. Second, we revisit previous work by Choudhury [Proc. AMS 108] who had provided a lower bound on $\alpha$ for preserving positivity when the blocks $H_{st}$ pairwise commute. We completely settle this problem by characterizing the full set of powers preserving positivity in this setting. Our characterizations generalize previous work by FitzGerald-Horn, Bhatia-Elsner, and Hiai from scalars to arbitrary block size, and in particular, generalize the Schur Product Theorem. Finally, a natural and unifying framework for studying the case of diagonalizable blocks consists of replacing real powers by general characters of the complex plane. We thus classify such characters, and generalize our results to this more general setting. In the course of our work, given $\beta\in\mathbb{Z}$, we provide lower and upper bounds for the threshold power $\alpha >0$ above which the complex characters $re^{i\theta}\mapsto r^\alpha e^{i\beta\theta}$ preserve positivity when applied entrywise to positive semidefinite matrices. In particular, we completely resolve the $n=3$ case of a question raised in 2001 by Xingzhi Zhan. As an application, we extend previous work by de Pillis [Duke Math. J. 36] by classifying the characters $K$ of the complex plane for which the map $(H_{st}){s,t=1}^n \mapsto (K({\rm tr}(H{st})))_{s,t=1}^n$ preserves positivity.