Some remarks on the projector associated with the intersection of two linear manifolds

  • Oscar E. Taurian
  • Per-Olov Löwdin
Atomic and Molecular Physics


The projector associated with the intersection of two linear manifolds is derived by studying the properties of the overlap matrix between the two sets spanning the linear manifolds and its adjoint matrix. The problem is reduced to an eigenvalue problem which — besides giving an explicit solution to the intersection problem — gives some additional information about the relationship between the two linear manifolds: for instance, the existence of “pairing regions” with partial overlap, and/or mutually orthogonal regions with zero overlap. Finally, an alternative approach is given which may be better suited for programming purposes.


Manifold Eigenvalue Problem Diagonal Matrice Linear Manifold Programming Purpose 
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  1. 1.
    See e.g.P. O. Löwdin, J. Appl. Phys.,33, 251, 1962; see alsoP. O. Löwdin, “Set Theory and Linear Algebra, Part I” (Florida Technical Note 490, 1976).MATHCrossRefADSGoogle Scholar
  2. 2.
    P. O. Löwdin, Phys. Rev.,139, A 357, 1965, particularly p. 370.CrossRefGoogle Scholar

Copyright information

© with the authors 1981

Authors and Affiliations

  • Oscar E. Taurian
    • 1
  • Per-Olov Löwdin
    • 2
  1. 1.Quantum Chemistry GroupUppsala UniversityUppsalaSweden
  2. 2.Also at Quantum Theory ProjectUniversity of FloridaGainesvilleUSA

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