  Entrywise powers of symmetric matrices preserving positivity, monotonicity or
convexity with respect to the Loewner ordering arise in various applications,
and have received much attention recently in the literature. Following
FitzGerald and Horn [J. Math. Anal. Appl., 1977], it is well-known that there
exists a critical exponent beyond which all entrywise powers preserve positive
definiteness. Similar phase transition phenomena have also recently been shown
by Hiai (2009) to occur for monotonicity and convexity. In this paper, we
complete the characterization of all the entrywise powers below and above the
critical exponents that are positive, monotone, or convex on the cone of
positive semidefinite matrices. We then extend the original problem by fully
classifying the positive, monotone, or convex powers in a more general setting
where additional rank constraints are imposed on the matrices. We also classify
the entrywise powers that are super/sub-additive with respect to the Loewner
ordering. Finally, we extend all the previous characterizations to matrices
with negative entries. Our analysis consequently allows us to answer a question
raised by Bhatia and Elsner (2007) regarding the smallest dimension for which
even extensions of the power functions do not preserve Loewner positivity.
