Integrally normalizable matrices with respect to a given set

The n×n matrix A is integrally normalizable with respect to a prescribed subset M of {(i,j):i,j=1,2,…,n and i≠j} provided A is diagonally similar to an integer matrix each of whose entries in positions corresponding to M is equal to 1. In the case that the elements of M form the arc set of a spanning tree, the matrices that are integrally normalizable with respect to M have been characterized. This paper gives a characterization for general subsets M. In addition, necessary and sufficient conditions for each matrix with a given zero-nonzero pattern to be integrally normalizable with respect to an arbitrary subset M are given.