  It is perhaps not widely recognized that certain common notions of distance
between probability measures have an alternative dual interpretation which
compares corresponding functionals against suitable families of test functions.
This dual viewpoint extends in a straightforward manner to suggest metrics
between matrix-valued measures. Our main interest has been in developing
weakly-continuous metrics that are suitable for comparing matrix-valued power
spectral density functions. To this end, and following the suggested recipe of
utilizing suitable families of test functions, we develop a weakly-continuous
metric that is analogous to the Wasserstein metric and applies to matrix-valued
densities. We use a numerical example to compare this metric to certain
standard alternatives including a different version of a matricial Wasserstein
metric developed earlier.
