  We give two different definitions of what it means for a matrix-valued
function to be log concave, guided by similar notions in complex differential
geometry. After discussing a few simple examples, we proceed to develop some of
the basic properties associated with these new concepts. Finally, we prove a
matrix-valued Pr\'ekopa theorem using a weighted, vector-valued Paley-Wiener
theorem, and positivity properties of direct image bundles.
