Fundamental Properties of the Box Product for Matrices over a Commutative Semiring and Johnson-Nylen Transformation

Abstract

We define the box product for matrices over an arbitrary commutative semiring, including the concepts of usual matrix product and Hadamard product. The box product possess the associativity, the identity, the distributivity over the addition, and the compatibility with the scalar multiplication and the transposition of a matrix. We investigate relationship between the box product and a block vector-operator. Moreover, we can transform the box product into the usual product via Johnson-Nylen transformation.