  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.
