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Linear transformations under which the doubly stochastic matrices are invariant


Author: Richard Sinkhorn
Journal: Proc. Amer. Math. Soc. 27 (1971), 213-221
MSC: Primary 15.65
DOI: https://doi.org/10.1090/S0002-9939-1971-0269678-1
MathSciNet review: 0269678
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Abstract: Let $ [{M_n}(C)]$ denote the set of linear maps from the $ n \times n$ complex matrices into themselves and let $ {\hat \Omega _n}$ denote the set of complex doubly stochastic matrices, i.e. complex matrices whose row and column sums are 1. If $ F \in [{M_n}(C)]$ is such that $ F({\hat \Omega _n}) \subseteq {\hat \Omega _n}$ and $ {F^ \ast }({\hat \Omega _n}) \subseteq {\hat \Omega _n}$, then there exist $ {A_i},{B_i},A$, and $ B \in {\hat \Omega _n}$ such that

$\displaystyle F(X) = \sum\limits_i {{A_i}X{B_i} + A{X^t}{J_n} + {J_n}{X^t}B - (1 + m){J_n}X{J_n}} $

for all $ n \times n$ complex matrices $ X$, where $ {J_n}$ is the $ n \times n$ matrix whose elements are each $ 1/n$ and where the superscript $ t$ denotes transpose. $ m$ denotes the number of the $ {A_i}$ (or $ {B_i}$).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0269678-1
Keywords: Faithful representation, generalized doubly stochastic matrices, Kroneker product, lexicographic ordering
Article copyright: © Copyright 1971 American Mathematical Society

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