Birkhoff factorization

In mathematics, Birkhoff factorization or Birkhoff decomposition, introduced by George David Birkhoff (1909), is a generalization of the LU decomposition (i.e. Gauss elimination) to loop groups.

The factorization of an invertible matrix ${\displaystyle M\in \mathrm {GL} _{n}(\mathbb {C} [z,z^{-1}])}$ with coefficients that are Laurent polynomials in ${\displaystyle z}$ is given by a product ${\displaystyle M=M^{+}M^{0}M^{-}}$, where ${\displaystyle M^{+}}$ has entries that are polynomials in ${\displaystyle z}$, ${\displaystyle M^{0}=\mathrm {diag} (z^{k_{1}},z^{k_{2}},...,z^{k_{n}})}$ is diagonal with ${\displaystyle k_{i}\in \mathbb {Z} }$ for ${\displaystyle 1\leq i\leq n}$ and ${\displaystyle k_{1}\geq k_{2}\geq ...\geq k_{n}}$, and ${\displaystyle M^{-}}$ has entries that are polynomials in ${\displaystyle z^{-1}}$. For a generic matrix we have ${\displaystyle M^{0}=\mathrm {id} }$.

Birkhoff factorization implies the Birkhoff–Grothendieck theorem of Grothendieck (1957) 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 Alexander Grothendieck (1957). 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.