On the Bhattacharya-Mesner rank of third order hypermatrices

We introduce the Bhattacharya-Mesner rank of third order hypermatrices as a relaxation to the tensor rank and devise from it some bounds for the tensor rank. We use the Bhattacharya-Mesner rank to extend to third order hypermatrices the connection relating the rank to a notion of linear dependence. We also derive explicit necessary and sufficient conditions for the existence of third order hypermatrix inverse pair. Finally we use inverse pair to extend to third order hypermatrices the formulation and proof of the matrix rank-nullity theorem.