Suppose that each person's utility depends on his or her own consumption as well as on the utilities of others. We consider the question of when a system of interdependent utility functions induces unique utility functions over allocations and identifies the class of transformations on interdependent utility functions that are equivalent in the sense of inducing the same preferences over allocations. We show that well-behaved systems of this kind can be studied by means of the theory of dominant-diagonal matrices and that the theory of dominant-diagonal matrices with finitely many elements extends in a satisfactory way to denumerable matrices. The theory of denumerable dominant diagonal matrices allows an elegant analysis of systems of intergenerational benevolence. We also revisit and extend the theory of two-sided altruism as formulated by Kimball and by Hori and Kanaya.