Abstract
A scaling of a non-negative, square matrixA ≠ 0 is a matrix of the formDAD −1, whereD is a non-negative, non-singular, diagonal, square matrix. For a non-negative, rectangular matrixB ≠ 0 we define a scaling to be a matrixCBE −1 whereC andE are non-negative, non-singular, diagonal, square matrices of the corresponding dimension. (For square matrices the latter definition allows more scalings.) A measure of the goodness of a scalingX is the maximal ratio of non-zero elements ofX. We characterize the minimal value of this measure over the set of all scalings of a given matrix. This is obtained in terms of cyclic products associated with a graph corresponding to the matrix. Our analysis is based on converting the scaling problem into a linear program. We then characterize the extreme points of the polytope which occurs in the linear program.
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This research was partially supported by National Science Foundation grants ENG 76-15599, MCS 76-06374 and MCS 78-01087 and a grant from the Graduate School of the University of Wisconsin. This work also relates to the Department of the Navy Contract No. ONR 00014-76-C-0085 issued under ONR contract authority NR 047-006.
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Rothblum, U.G., Schneider, H. Characterizations of optimal scalings of matrices. Mathematical Programming 19, 121–136 (1980). https://doi.org/10.1007/BF01581636
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DOI: https://doi.org/10.1007/BF01581636