An Axiomatic Approach to Proportionality between Matrices

Given a matrix p ≥ 0 what does it mean to say that a matrix f (of the same dimension), whose row and column sums must fall between specific limits, is "proportional to" p? This paper gives an axiomatic solution to this question in two distinct contexts. First, for any real "allocation" matrix f. Second, for any integer constrained "apportionment" matrix f. In the case of f real the solution turns out to coincide with what has been variously called biproportional scaling and diagonal equivalence and has been much used in econometrics and statistics. In the case of f integer the problem arises in the simultaneous apportionment of seats to regions and to parties and also in the rounding of tables of census data.