Abstract
A scaling of a nonnegative matrixA is a matrixXAY −1, whereX andY are nonsingular, nonnegative diagonal matrices. Some condition may be imposed on the scaling, for example, whenA is square,X=Y or detX=detY. We characterize matrices for which there exists a scaling that satisfies predetermined upper and lower bound. Our principal tools are a piecewise linear theorem of the alternative and a theorem decomposing a solution of a system of equations as a sum of minimal support solutions which conform with the given solutions.
Similar content being viewed by others
References
M. Bacharach,Biproportional matrices and input-output change (Cambridge University Press, Cambridge, 1970).
A. Bachem and M. Grötschel, “New Aspects of polyhedral theory”, in: B. Korte, ed.,Modern applied mathematics (Optimization and operations research) (North-Holland, Amsterdam, 1982) pp. 51–106.
R.E. Bank, “An automatic scaling procedure for a D'Yakanov-Gunn interation scheme”,Linear Algebra and its Applications 28 (1979) 17–33.
F.L. Bauer, “Optimally scaled matrices”,Numerische Mathematik 5 (1963) 73–87.
F.L. Bauer, “Remarks on optimally scaled matrices”,Numerische Mathematik 13 (1969), 1–3.
C. Berge,Graphs and hypergraphs (North-Holland, Amsterdam, 1973).
R.G. Bland, “Complementary orthogonal subspaces of ℝn and orientability of matroids”, Ph.D. thesis, available as Technical Report No. 219, (Cornell University, Ithaca, New York, 1974).
R.G. Bland and M. Las Vergnas, “Minty coloring and orientations of matroids”, in: A. Gewirtz and L.v. Quintas, eds.,Proceedings of the second international conference on combinatories mathematics (Annals of the New York Academy of Sciences 319, 1979).
A.R. Curtis and J.K. Reid, “On the automatic scaling of matrices for Gaussian elimination”,Journal of the Institute of Mathematics and its Applications 10 (1972) 118–124.
G.M. Engel and H. Schneider, “Cyclic and diagonal products on a matrix”,Linear Algebra and its Applications 7 (1973) 301–335.
G.M. Engel and H. Schneider, “Algorithms for testing the diagonal similarity of matrices and related problems”,SIAM Journal of Algebraic and Discrete Mathematics 3 (1982), 429–438.
M. Fiedler and V. Ptak, “Cyclic products and an inequality for determinants”,Czechoslovak Mathematical Journal 19 (1969) 428–450.
D.R. Fulkerson and P. Wolfe, “An algorithm for scaling matrices”,SIAM Review 4 (1962) 142–146.
D. Gale,The theory of linear economic models (McGraw-Hill, New York, 1966).
A.J. Goldman, “Resolution and separation theorems for polyhedral convex sets”, in: H.W. Kuhn and A.W. Tucker, eds.,Linear inequalities and related systems (Princeton University Press 83, 1956).
M.v. Golitschek, “An algorithm for scaling matrices and computing the minimum cycle mean in a digraph”,Numerische Mathematik 35 (1980) 45–55.
M.v Golitschek, “Optimal cycles in doubly weighted graphs and approximation of bivariate functions by univariate ones”,Numerische Mathematik 39 (1982) 65–84.
R.W. Hamming,Introduction to applied numerical analysis (McGraw-Hill, New York, 1971).
T.S. Motzkin, “Beiträge zur Theorie der linearen Ungleichungen”, Univ. Basel Dissertation (Jerusalem, Israel, 1936).
W. Orchard-Hays,Advanced linear programming computing techniques (McGraw-Hill, New York, 1968).
R.T. Rockafellar, “The elementary vectors of a subspace of ℝn”, in: R.C. Bose and T.A. Dowling, eds.,Combinatorial mathematics and its applications, Proc. of the Chapel Hill Conference (University of North Carolina Press, 1968) pp. 104–127.
U.G. Rothblum and H. Schneider, “Flows on graphs applied to diagonal similarity and diagonal equivalence for matrices”,Discrete Mathematics 24 (1978) 202–220.
B.D. Saunders and H. Schneider, “Cones, graphs and optimal scalings of matrices”Linear and Multilinear Algebra 8 (1979) 121–135.
J.A. Tomlin, “On scaling linear programming problems”,Mathematical Programming Study 4 (1975) 146–166.
W.T. Tutte, “A class of Abelian groups”,Canadian Journal of Mathematics 8 (1956) 13–28.
Author information
Authors and Affiliations
Additional information
This research was partially supported by National Science Foundation Grants ENG-78-25182 and MCS-80-26132.
Rights and permissions
About this article
Cite this article
v. Golitschek, M., Rothblum, U.G. & Schneider, H. A conforming decomposition theorem, a piecewise linear theorem of the alternative, and scalings of matrices satisfying lower and upper bounds. Mathematical Programming 27, 291–306 (1983). https://doi.org/10.1007/BF02591905
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02591905