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 s ...

Birkhoff Factorization
In mathematics, Birkhoff factorization or Birkhoff decomposition, introduced by , is a generalization of the LU decomposition (i.e. Gauss elimination) to loop groups. The factorization of an invertible matrix M\in\mathrm_n(\mathbb ,z^ with coefficients that are Laurent polynomials in z is given by a product M=M^M^M^, where M^ has entries that are polynomials in z, M^=\mathrm(z^, z^,...,z^) is diagonal with k_i\in\mathbb for 1\leq i\leq n and k_1\geq k_2\geq ...\geq k_n, and M^ has entries that are polynomials in z^. For a generic matrix we have M^=\mathrm. Birkhoff factorization implies the Birkhoff–Grothendieck theorem of that vector bundles over the projective line are sums of line bundles. There are several variations where the general linear group is replaced by some other reductive algebraic group, due to . Birkhoff factorization follows from the Bruhat decomposition for affine Kac–Moody groups (or loop groups), and conversely the Bruhat decomposition for the affine g ...
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