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On uniqueness, analyticity, and first- and second-order derivatives of Sinkhorn-type DAD scalings
Citation Link: https://doi.org/10.15480/882.16817
Publikationstyp
Journal Article
Date Issued
2026-05-15
Sprache
English
Author(s)
TORE-DOI
Volume
737
Start Page
173
End Page
192
Citation
Linear Algebra and Its Applications 173: 173-192 (2026)
Publisher DOI
Scopus ID
Publisher
Elsevier
We call a real square matrix A scalable if there is a diagonal matrix D with positive diagonal entries such that all row sums of D A D are equal to 1. In this case, Dis called a scaling for A. For an entrywise nonnegative A, this means that D A D is a stochastic matrix. We prove that such a nonnegative A has a unique scaling if and only if it has a scaling Dsuch that −1 is not an eigenvalue of D A D. If, on the contrary, a nonnegative A has multiple scalings, then we show that it already has infifinitely many scalings. Furthermore, we prove that the function which maps a uniquely scalable, nonnegative matrix Ato the diagonal vector x = x(A)of its scaling, which we call the Sinkhorn vector of A, is a real analytic function. Finally, we give explicit, index-free formulas for the Jacobian and Hessian matrices of xwith respect to the entries of A. In particular, we prove that ∂xi/∂Ai,j ≤ −1/2xi²xj < 0 for all i, j ∈ {1,...,n} where n is the order of A.
Subjects
Positive diagonal congruence
Positive matrices
Sinkhorn's DAD theorem
Stochastic matrices
DDC Class
510: Mathematics
Publication version
publishedVersion
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1-s2.0-S002437952600056X-main.pdf
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994.53 KB
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