In mathematics, Birkhoff factorization or Birkhoff decomposition, introduced by , is the factorization of an
invertible matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
''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 In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a faithf ...
is replaced by some other reductive algebraic group, due to .
Birkhoff factorization implies the
Birkhoff–Grothendieck theorem
In mathematics, the Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over \mathbb^1 is a direct sum of holomorphic line bundles. The theorem was ...
of that
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 (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
s over the projective line are sums of
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 organisin ...
s.
Birkhoff factorization follows from the
Bruhat decomposition In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) ''G'' = ''BWB'' of certain algebraic groups ''G'' into cells can be regarded as a general expression of the principle ...
for affine Kac–Moody groups (or
loop group
In mathematics, a loop group is a Group (mathematics), group of Loop (topology), loops in a topological group ''G'' with multiplication defined pointwise.
Definition
In its most general form a loop group is a group of continuous mappings from a ...
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.
See also
*
Birkhoff decomposition (disambiguation) 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 ...
*
Riemann–Hilbert problem
In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems h ...
References
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Matrices
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