We consider the problem of taking a matrix A and finding diagonal matrices D and E such that the rows and columns of B = DAE satisfy some specific constraints. Examples of constraints are that
* the row and column sums of B should all equal one;
* the norms of the rows and columns of B should all be equal;
* the row and column sums of B should take values specified by vectors p and q.
Simple iterative algorithms for solving these problems have been known for nearly a century. We provide a simple framework for describing these algorithms that allow us to develop robust convergence results and describe a straightforward approach to accelerate the rate of convergence.
We describe some of the diverse applications of balancing with examples from preconditioning, clustering, network analysis and psephology.
This is joint work with Kerem Akartunali (Strathclyde), Daniel Ruiz (ENSEEIHT, Toulouse) and Bora Ucar (ENS, Lyon).