  Some natural inequalities related to rearrangement in matrix products can
also be regarded as extensions of classical inequalities for sequences or
integrals. In particular, we show matrix versions of Chebyshev and Kantorovich
type inequalities. The matrix approach may also provide simplified proofs and
new results for classical inequalities. For instance, we show a link between
Cassel's inequality and the basic rearrangement inequality for sequences of
Hardy-Littlewood-Polya, and we state a reverse inequality to the
Hardy-Littlewood-Polya inequality in which matrix technics are essential.
