  We propose a contemporaneous bilinear transformation for matrix time series
to alleviate the difficulties in modelling and forecasting large number of time
series together. More precisely the transformed matrix splits into several
small matrices, and those small matrix series are uncorrelated across all
times. Hence an effective dimension-reduction is achieved by modelling each of
those small matrix series separately without the loss of information on the
overall linear dynamics. We adopt the bilinear transformation such that the
rows and the columns of the matrix do not mix together, as they typically
represent radically different features. As the targeted transformation is not
unique, we identify an ideal version through a new normalization, which
facilitates the no-cancellation accumulation of the information from different
time lags. The non-asymptotic error bounds of the estimated transformation are
derived, leading to the uniform convergence rates of the estimation. The
proposed method is illustrated numerically via both simulated and real data
examples.
