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Birkhoff decomposition refers to two different mathematical concepts:
The Birkhoff factorization, introduced by George David Birkhoff at 1909, is the presentation of an invertible matrix with polynomial coefficients as a product of three matrices.
The Birkhoff - von Neumann decomposition, introduced by Garrett Birkhoff (George's son) at 1946, is the presentation of a bistochastic matrix as a convex sum of permutation matrices. It can be found by the Birkhoff algorithm.
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This disambiguation page lists articles associated with the title Birkhoff decomposition.