  Subdifferentials (in the sense of convex analysis) of matrix-valued functions
defined on $\mathbb{R}^d$ that are convex with respect to the L\"{o}wner
partial order can have a complicated structure and might be very difficult to
compute even in simple cases. The aim of this paper is to study subdifferential
calculus for such functions and properties of their subdifferentials. We show
that many standard results from convex analysis no longer hold true in the
matrix-valued case. For example, in this case the subdifferential of the sum is
not equal to the sum of subdifferentials, the Clarke subdifferential is not
equal to the subdifferential in the sense of convex analysis, etc. Nonetheless,
it is possible to provide simple rules for computing nonempty subsets of
subdifferentials (in particular, individual subgradients) of convex
matrix-valued functions in the general case and to completely describe
subdifferentials of such functions defined on the real line. As a by-product of
our analysis, we derive some interesting properties of convex matrix-valued
functions, e.g. we show that if such function is nonsmooth, then its diagonal
elements must be nonsmooth as well.
