  We propose a contemporaneous bilinear transformation for a $p\times q$ matrix
time series to alleviate the difficulties in modeling and forecasting matrix
time series when $p$ and/or $q$ are large. The resulting transformed matrix
assumes a block structure consisting of several small matrices, and those small
matrix series are uncorrelated across all times. Hence an overall parsimonious
model is achieved by modelling each of those small matrix series separately
without the loss of information on the linear dynamics. Such a parsimonious
model often has better forecasting performance, even when the underlying true
dynamics deviates from the assumed uncorrelated block structure after
transformation. The uniform convergence rates of the estimated transformation
are derived, which vindicate an important virtue of the proposed bilinear
transformation, i.e. it is technically equivalent to the decorrelation of a
vector time series of dimension max$(p,q)$ instead of $p\times q$. The proposed
method is illustrated numerically via both simulated and real data examples.
