Products of multidimensional matrices, stochastic matrices, and permanents

Abstract

The paper is devoted to four basic multidimensional matrix operations (outer product, Kronecker product,
contraction, and projection) and two derivative operations (dot and circle products). It is studied
the interrelations between these operations, some of their algebraic properties, and their action
on k-stochastic matrices. Also, it is proved several relations on the permanents of products of
multidimensional matrices. In particular, it is shown that the permanent of the dot product of
nonnegative multidimensional matrices is not less than the product of their permanents. Another result
of the paper is that inequalities on the Kronecker product of nonnegative 2-dimensional matrices
cannot be extended to the multidimensional case.