  Doubly non-negative matrices arise naturally in many setting including Markov
random fields (positively banded graphical models) and in the convergence
analysis of Markov chains. In this short note, we settle a recent conjecture by
C.R. Johnson et al. [Linear Algebra Appl. 435 (2011)] by proving that the
critical exponent beyond which all continuous conventional powers of $n$-by-$n$
doubly nonnegative matrices are doubly nonnegative is exactly $n-2$. We show
that the conjecture follows immediately by applying a general characterization
from the literature. We prove a stronger form of the conjecture by classifying
all powers preserving doubly nonnegative matrices, and proceed to generalize
the conjecture for broad classes of functions. We also provide different
approaches for settling the original conjecture.
