  In this paper, we fully characterize the duality mapping over the space of
matrices that are equipped with Schatten norms. Our approach is based on the
analysis of the saturation of the H\"older inequality for Schatten norms. We
prove in our main result that, for $p\in (1,\infty)$, the duality mapping over
the space of real-valued matrices with Schatten-$p$ norm is a continuous and
single-valued function and provide an explicit form for its computation. For
the special case $p = 1$, the mapping is set-valued; by adding a rank
constraint, we show that it can be reduced to a Borel-measurable single-valued
function for which we also provide a closed-form expression.
