  In this article, we first propose generalized row/column matrix Kendall's tau
for matrix-variate observations that are ubiquitous in areas such as finance
and medical imaging. For a random matrix following a matrix-variate
elliptically contoured distribution, we show that the eigenspaces of the
proposed row/column matrix Kendall's tau coincide with those of the row/column
scatter matrix respectively, with the same descending order of the eigenvalues.
We perform eigenvalue decomposition to the generalized row/column matrix
Kendall's tau for recovering the loading spaces of the matrix factor model. We
also propose to estimate the pair of the factor numbers by exploiting the
eigenvalue-ratios of the row/column matrix Kendall's tau. Theoretically, we
derive the convergence rates of the estimators for loading spaces, factor
scores and common components, and prove the consistency of the estimators for
the factor numbers without any moment constraints on the idiosyncratic errors.
Thorough simulation studies are conducted to show the higher degree of
robustness of the proposed estimators over the existing ones. Analysis of a
financial dataset of asset returns and a medical imaging dataset associated
with COVID-19 illustrate the empirical usefulness of the proposed method.
