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Birkhoff factorization

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In mathematics, Birkhoff factorization or Birkhoff decomposition, introduced by Birkhoff (1909), is the factorization of a 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.

Birkhoff factorization implies the Birkhoff–Grothendieck theorem of Grothendieck (1957) that vector bundles over the projective line are sums of line bundles.

References

  • Birkhoff, George David (1909), "Singular points of ordinary linear differential equations", Transactions of the American Mathematical Society, 10 (4): 436–470, ISSN 0002-9947, JFM 40.0352.02
  • Grothendieck, Alexander (1957), "Sur la classification des fibrés holomorphes sur la sphère de Riemann", American Journal of Mathematics, 79: 121–138, ISSN 0002-9327, MR0087176
  • Khimshiashvili, G. (2001) [1994], "Birkhoff factorization", Encyclopedia of Mathematics, EMS Press
  • Pressley, Andrew; Segal, Graeme (1986), Loop groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, ISBN 978-0-19-853535-5, MR900587
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