  Matrix-variate data of high dimensions are frequently observed in finance and
economics, spanning extended time periods, such as the long-term data on
international trade flows among numerous countries. To address potential
structural shifts and explore the matrix structure's informational context, we
propose a time-varying matrix factor model. This model accommodates changing
factor loadings over time, revealing the underlying dynamic structure through
nonparametric principal component analysis and facilitating dimension
reduction. We establish the consistency and asymptotic normality of our
estimators under general conditions that allow for weak correlations across
time, rows, or columns of the noise. A novel approach is introduced to overcome
rotational ambiguity in the estimators, enhancing the clarity and
interpretability of the estimated loading matrices. Our simulation study
highlights the merits of the proposed estimators and the effective of the
smoothing operation. In an application to international trade flow, we
investigate the trading hubs, centrality, patterns, and trends in the trading
network.
