Right Semi-Tensor Product for Matrices Over a Commutative Semiring

Abstract

This paper generalizes the right semi-tensor product for real matrices to that for matrices over an arbitrary commutative semiring, and investigates its properties. This product is defined for any pair of matrices satisfying the matching-dimension condition. In particular, the usual matrix product and the scalar multiplication are its special cases. The right semi-tensor product turns out to be an associative bilinear map that is compatible with the transposition and the inversion. The product also satisfies certain identity-like properties and preserves some structural properties of matrices. We can convert between the right semi-tensor product of two matrices and the left semi-tensor product using commutation matrices. Moreover, certain vectorizations of the usual product of matrices can be written in terms of the right semi-tensor product.