On scaling to an integer matrix and graphs with integer weighted cycles

Between 1970 and 1982 Hans Schneider and co-authors produced a number of results regarding matrix scalings. They demonstrated that a matrix has a diagonal similarity scaling to any matrix with entries in the subgroup generated by the cycle weights of the associated digraph, and that a matrix has an equivalent scaling to any matrix with entries related to the weights of cycles in an associated bipartite graph. Further, given matrices A and B, they produced a description of all diagonal X such that X−1AX=B. In 2005 Butkovič and Schneider used max-algebra to give a simple and efficient description of this set of scalings. In this paper we focus on the additive group of integers, and work in the max-plus algebra to give a full description of all scalings of a real matrix A to any integer matrix. We do this for four types of scalings; beginning with the familiar X−1AX, XAY and XAX scalings and finishing with a new scaling which we call a signed similarity scaling. This is a scaling of the form XAY where we specify for each row i, either xi=yi or xi=−yi. In all of our results we use necessary and sufficient conditions for existence which are based on integer weighted cycles in the associated digraph, or associated bipartite graph, of the matrix.