  Modern technology often generates data with complex structures in which both
response and explanatory variables are matrix-valued. Existing methods in the
literature are able to tackle matrix-valued predictors but are rather limited
for matrix-valued responses. In this article, we study matrix-variate
regressions for such data, where the response Y on each experimental unit is a
random matrix and the predictor X can be either a scalar, a vector, or a
matrix, treated as non-stochastic in terms of the conditional distribution Y|X.
We propose models for matrix-variate regressions and then develop envelope
extensions of these models. Under the envelope framework, redundant variation
can be eliminated in estimation and the number of parameters can be notably
reduced when the matrix-variate dimension is large, possibly resulting in
significant gains in efficiency. The proposed methods are applicable to high
dimensional settings.
