  Vector Fitting (VF) is a popular method of constructing rational approximants
that provides a least squares fit to frequency response measurements. In an
earlier work, we provided an analysis of VF for scalar-valued rational
functions and established a connection with optimal $H_2$ approximation. We
build on this work and extend the previous framework to include the
construction of effective rational approximations to matrix-valued functions, a
problem which presents significant challenges that do not appear in the scalar
case. Transfer functions associated with multi-input/multi-output (MIMO)
dynamical systems typify the class of functions that we consider here. Others
have also considered extensions of VF to matrix-valued functions and related
numerical implementations are readily available. However to our knowledge, a
detailed analysis of numerical issues that arise does not yet exist. We offer
such an analysis including critical implementation details here.
  One important issue that arises for VF on matrix-valued functions that has
remained largely unaddressed is the control of the McMillan degree of the
resulting rational approximant; the McMillan degree can grow very high in the
case of large input/output dimensions. We introduce two new mechanisms for
controlling the McMillan degree of the final approximant, one based on
alternating least-squares minimization and one based on ancillary
system-theoretic reduction methods. Motivated in part by our earlier work on
the scalar VF problem as well as by recent innovations for computing optimal
$H_2$ approximation, we establish a connection with optimal $H_2$
approximation, and are able to improve significantly the fidelity of VF through
numerical quadrature, with virtually no increase in cost or complexity. We
provide several numerical examples to support the theoretical discussion and
proposed algorithms.
