Abstract
LetR be the set of nonnegative matrices whose row and column sums fall between specific limits and whose entries sum to some fixedh > 0. Closely related axiomatic approaches have been developed to ascribe meanings to the statements: the real matrixf ∈ R and the integer matrixa ∈ R are “proportional to” a given matrixp ≥ 0.
These approaches are described, conditions under which proportional solutions exist are characterized, and algorithms are given for finding proportional solutions in each case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.M. Anthonisse, “Proportional representation in a regional council,” CWI Newsletter 5, Centrum voor Wiskunde en Informatica (Amsterdam, December 1984) 22–29.
M. Bacharach,Biproportional Matrices and Input-Output Change (Cambridge University Press, Cambridge, 1970).
M.L. Balinski and G. Demange, “An axiomatic approach to proportionality between matrices,”Mathematics of Operations Research (1989), to appear.
M.L. Balinski and H.P. Young,Fair Representation: Meeting the Ideal of One Man, One Vote (Yale University Press, New Haven, CT, 1982).
J. Bigelow and N. Shapiro, “A scaling theorem and algorithms,”SIAM Journal of Applied Mathematics 33 (1977) 348–352.
L.H. Cox and L.R. Ernst, “Controlled rounding,”INFOR 20 (1982) 423–432.
D.G. Luenberger,Introduction to Linear and Nonlinear Programming (Addison-Wesley, Reading, MA, 1973).
R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
R. Sinkhorn, “Diagonal equivalence to matrices with prescribed row and column sums,”American Mathematical Monthly 74 (1967) 402–405.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Balinski, M.L., Demange, G. Algorithms for proportional matrices in reals and integers. Mathematical Programming 45, 193–210 (1989). https://doi.org/10.1007/BF01589103
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01589103