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.
Topics referred to by the same term
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Birkhoffdecomposition refers to two different mathematical concepts: The Birkhoff factorization, introduced by George David Birkhoff at 1909, is the...
Birkhoff's algorithm (also called Birkhoff-von-Neumann algorithm) is an algorithm for decomposing a bistochastic matrix into a convex combination of permutation...
In mathematics, Birkhoff factorization or Birkhoffdecomposition, introduced by George David Birkhoff (1909), is the factorization of an invertible matrix...
theorem is given below. This representation is known as the Birkhoff–von Neumann decomposition, and may not be unique. It is often described as a real-valued...
be considered as a special case of the Bruhat decomposition. The Birkhoffdecomposition, a special case of the Bruhat decomposition for affine groups....
to the space average. Two of the most important theorems are those of Birkhoff (1931) and von Neumann which assert the existence of a time average along...
W}(Bw_{1}B\cap B_{-}w_{2}B_{-}).} Lie group decompositionsBirkhoff factorization, a special case of the Bruhat decomposition for affine groups. Cluster algebra...
composition series, but not transfinite descending composition series (Birkhoff 1934). Baumslag (2006) gives a short proof of the Jordan–Hölder theorem...
invariant Poincaré–Birkhoff–Witt theorem, usually known as the PBW theorem Shirshov–Witt theorem Witt algebra Witt decomposition Witt design (Witt geometry)...
paper by von Neumann and Garrett Birkhoff, the first to introduce quantum logics, wherein von Neumann and Birkhoff first proved that quantum mechanics...
Adkins&Weintraub (1992) p.272 Rudin 1991, pp. 306–312. "Orthogonal Complement" G. D. Birkhoff (1923) Relativity and Modern Physics, pages 62,63, Harvard University Press...
{\displaystyle \mathbb {CP} ^{n}} , with decomposition of the same type. Lines such that the decomposition differs from this generic type are called...
{\displaystyle r>0} , are pre-compact. The Birkhoff–Kakutani theorem (named after mathematicians Garrett Birkhoff and Shizuo Kakutani) states that the following...
form Frobenius normal form Jordan matrix Jordan–Chevalley decomposition Matrix decomposition Modal matrix Weyr canonical form Shilov defines the term Jordan...
However, the decomposition is not unique, and some decompositions may be better than others. Budish, Che, Kojima and Milgrom generalize Birkhoff's algorithm...
Birkhoff metrized the positive cone using Hilbert's projective metric and proved Jentsch's theorem using the contraction mapping theorem. Birkhoff's results...
{\displaystyle M} is dense in C ( M ) {\displaystyle C(M)} . In 1932 George Birkhoff described his "remarkable closed curve", a homeomorphism of the annulus...
Lane & Birkhoff (1999). A proof of this can be found in more generality in Bourbaki (1989). See Bourbaki (1989, §III.7.1), and Mac Lane & Birkhoff (1999...
Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (also known as splitting extension). Given a...
dissection was introduced by George (1973); the name was suggested by Garrett Birkhoff. Nested dissection consists of the following steps: Form an undirected...
semialgebraic mappings. Piecewise polynomial mappings (see the Pierce–Birkhoff conjecture) are also semialgebraic mappings. Computational real algebraic...
the direct product of G and H is also called a direct sum (Mac Lane & Birkhoff 1999, §V.6). Thus the Cartesian product G × H is equipped with the structure...
The formal system takes as its starting point an observation of Garrett Birkhoff and John von Neumann, that the structure of experimental tests in classical...
isomorphisms will not commute with, say, the zero map; see (Mac Lane & Birkhoff 1999, §VI.4) for detailed discussion. Starting from finite-dimensional...
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