The ±-rank of a (0,±1)-matrix

We introduce and study a new rank of ( 0 , ± 1 ) -matrices generalizing both the binary rank and the term rank of ( 0 , 1 ) -matrices. We call it the ± -rank of a ( 0 , ± 1 ) -matrix. We establish several inequalities relating the different ranks, including ordinary real rank. Moreover, the ±-rank is discussed for certain classes of ( 0 , ± 1 ) -matrices, such as alternating sign matrices (ASMs), network matrices, and matrices whose bipartite graph is a tree. Computationally the ±-rank is not easy to determine (like binary rank it may be an NP-hard problem), and we investigate several examples.