Abstract
A square matrix is called line-sum-symmetric if the sum of elements in each of its rows equals the sum of elements in the corresponding column. Let A be an n×n nonnegative matrix and let X and Y be n×n diagonal matrices having positive diagonal elements. Then the matrices XA, XAX−1 and XAY are called a row-scaling, a similarity-scaling and an equivalence-scaling of A. The purpose of this paper is to study the different forms of line-sum-symmetric scalings of square nonnegative matrices. In particular, we characterize matrices for which such scalings exist and show uniqueness of similarity-scalings and uniqueness of row-scalings, up to a scalar multiple of the blocks corresponding to the classes of the given matrix.
This research was partially supported by National Science Foundation Grants MCS-81-21838, ECS-83-10213, MCS-80-26132 and DMS-8320189.
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Dedicated to G.B. Dantzig on the occasion of his seventieth birthday.
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© 1985 The Mathematical Programming Society, Inc.
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Eaves, B.C., Hoffman, A.J., Rothblum, U.G., Schneider, H. (1985). Line-sum-symmetric scalings of square nonnegative matrices. In: Cottle, R.W. (eds) Mathematical Programming Essays in Honor of George B. Dantzig Part II. Mathematical Programming Studies, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121080
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DOI: https://doi.org/10.1007/BFb0121080
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