Birkhoff Decomposition (other)
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 Birkhoff's algorithm (also called Birkhoff-von-Neumann algorithm) is an algorithm for decomposing a bistochastic matrix into a convex combination of permutation matrices. It was published by Garrett Birkhoff in 1946. It has many applications. One su ...

Birkhoff Factorization
In mathematics, Birkhoff factorization or Birkhoff decomposition, introduced by , is the factorization of an invertible matrix ''M'' with coefficients that are Laurent polynomials in ''z'' into a product ''M'' = ''M''+''M''0''M''−, where ''M''+ has entries that are polynomials in ''z'', ''M''0 is diagonal, and ''M''− has entries that are polynomials in ''z''−1. There are several variations where the general linear group is replaced by some other reductive algebraic group, due to . Birkhoff factorization implies the Birkhoff–Grothendieck theorem of that vector bundles over the projective line are sums of line bundles. Birkhoff factorization follows from the Bruhat decomposition for affine Kac–Moody groups (or loop groups), and conversely the Bruhat decomposition for the affine general linear group follows from Birkhoff factorization together with the Bruhat decomposition for the ordinary general linear group. See also *Birkhoff decomposition (disamb ...
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