  This paper introduces a matrix quantile factor model for matrix-valued data
with a low-rank structure. We estimate the row and column factor spaces via
minimizing the empirical check loss function over all panels. We show the
estimates converge at rate $1/\min\{\sqrt{p_1p_2}, \sqrt{p_2T},$
$\sqrt{p_1T}\}$ in average Frobenius norm, where $p_1$, $p_2$ and $T$ are the
row dimensionality, column dimensionality and length of the matrix sequence.
This rate is faster than that of the quantile estimates via ``flattening" the
matrix model into a large vector model. Smoothed estimates are given and their
central limit theorems are derived under some mild condition. We provide three
consistent criteria to determine the pair of row and column factor numbers.
Extensive simulation studies and an empirical study justify our theory.
