Right Semi-Tensor Product for Matrices Over a Commutative Semiring
DOI:
https://doi.org/10.26713/jims.v12i1.1217Keywords:
Right semi-tensor product, Kronecker product, Commutative semiring, Vector operator, Commutation matrixAbstract
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.
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