Equisum matrices and their permanence

In a square equisum matrix, all row and column sums are equal. In a rectang ular equisum matrix, the common row sum is a rational multiple of the commo n column sum. This paper explores properties of equisum matrices, in partic ular, the preservation of the equisum condition under a variety of linear, nonlinear and pattern-maintaining transformations. A principal tool employe d is a representation via the Fourier matrix or the circulant projectors as sociated with it.