  In this article, we introduce a notion of an exponential matrix, which is a
polynomial matrix with exponential properties, and a notion of an equivalence
relation of two exponential matrices, and then we initiate to study classifying
exponential matrices in positive characteristic, up to equivalence. We classify
exponential matrices of Heisenberg groups in positive characteristic, up to
equivalence. We also classify exponential matrices of size four-by-four in
positive characteristic, up to equivalence. From these classifications, we
obtain a classification of modular representations of elementary abelian
$p$-groups into Heisenberg groups, up to equivalence, and a classification of
four-dimensional modular representations of elementary abelian $p$-groups, up
to equivalence.
