In mathematics, Birkhoff factorization or Birkhoff decomposition, introduced by George David Birkhoff (1909), 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 Alexander Grothendieck (1957).
Birkhoff factorization implies the Birkhoff–Grothendieck theorem of Grothendieck (1957) 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.
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In mathematics, Birkhofffactorization or Birkhoff decomposition, introduced by George David Birkhoff (1909), is the factorization of an invertible matrix...
Birkhoff decomposition refers to two different mathematical concepts: The Birkhofffactorization, introduced by George David Birkhoff at 1909, is the...
w_{2}\in W}(Bw_{1}B\cap B_{-}w_{2}B_{-}).} Lie group decompositions Birkhofffactorization, a special case of the Bruhat decomposition for affine groups. Cluster...
essentially unique factorization into prime or irreducible factors. When the coefficient ring is a field or other unique factorization domain, an irreducible...
renormalization in quantum field theory. Renormalization was interpreted as Birkhofffactorization of loops in the character group of the associated Hopf algebra....
suffix of the given string. A factorization into a nonincreasing sequence of Lyndon words (the so-called Lyndon factorization) can be constructed in linear...
In a unique factorization domain (or more generally, a GCD domain), an irreducible element is a prime element. While unique factorization does not hold...
and to make clear the ramifications of the parallel postulate. Birkhoff's axioms: Birkhoff proposed four postulates for Euclidean geometry that can be confirmed...
statistical theory, particularly in multivariate analysis. Beurling's factorization theorem and much of the work on (abstract) harmonic analysis sought...
of N is denoted by Z; its elements are called the rational integers.] Birkhoff, Garrett (1948). Lattice Theory (Revised ed.). American Mathematical Society...
2: Greatest common divisor, pp. 856–862. Saunders Mac Lane and Garrett Birkhoff. A Survey of Modern Algebra, Fourth Edition. MacMillan Publishing Co.,...
and can be used to perform the commutations required by the Poincaré–Birkhoff–Witt theorem used in the construction of a universal enveloping algebra...
composition series, but not transfinite descending composition series (Birkhoff 1934). Baumslag (2006) gives a short proof of the Jordan–Hölder theorem...
mathematical theory of lattices and ordered sets that was developed by Garrett Birkhoff and others in the 1930s. Formal concept analysis finds practical application...
Categories (8): 1–24, MR 0948965 Lawvere, F. William (February 2016), "Birkhoff's Theorem from a geometric perspective: A simple example", Categories and...
1-factorable. The perfect 1-factorization conjecture that every complete graph on an even number of vertices admits a perfect 1-factorization. Cereceda's conjecture...
ax + by = c Integer factorization: breaking an integer into its prime factors Congruence of squares Dixon's algorithm Fermat's factorization method General...
Cholesky factorization — sparse approximation to the Cholesky factorization Incomplete LU factorization — sparse approximation to the LU factorization Uzawa...
It can be considered as a special case of the Bruhat decomposition. The Birkhoff decomposition, a special case of the Bruhat decomposition for affine groups...
Concrete Categories (PDF). Wiley. ISBN 0-471-60922-6. Mac Lane, Saunders; Birkhoff, Garrett (1999). Algebra (3rd ed.). American Mathematical Society. ISBN 0-8218-1646-2...
Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248. Birkhoff, George D. (1913). "Note on the gamma function". Bull. Amer. Math. Soc...
dimension (m+nm)=(m+nn){\textstyle {\binom {m+n}{m}}={\binom {m+n}{n}}}). The Birkhoff-Grothendieck theorem states that on the projective line, any vector bundle...
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