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⁺M⁰M⁻, where M⁺ has entries that are polynomials in z, M⁰ 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 bundle

Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

s over the projective line are sums of line bundle

Line bundle

In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...

s.

Birkhoff factorization follows from the Bruhat decomposition

Bruhat decomposition

In mathematics, the Bruhat decomposition G = BWB into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases...

 for affine algebraic groups (or loop group

Loop group

In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise. Specifically, letLG \,denote the space of continuous mapsS^1 \to G...

s), 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.