Poincaré separation theorem

E825436

mathematical theorem result in linear algebra result in spectral theory

The Poincaré separation theorem is a result in linear algebra and spectral theory that characterizes how the eigenvalues of a symmetric matrix relate to those of its principal submatrices, closely connected to eigenvalue interlacing phenomena.

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Statements (40)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in linear algebra ⓘ result in spectral theory ⓘ
appearsIn	texts on matrix analysis ⓘ texts on numerical linear algebra ⓘ
appliesTo	Hermitian matrices NERFINISHED ⓘ real symmetric matrices ⓘ
appliesWhen	considering invariant subspaces and their orthogonal complements ⓘ
assumes	eigenvalues are ordered nonincreasingly or nondecreasingly ⓘ matrix is symmetric or Hermitian ⓘ
characterizes	how eigenvalues change when restricting a symmetric operator to a subspace ⓘ spectral separation between a matrix and its compressions ⓘ
concerns	eigenvalue interlacing ⓘ eigenvalues ⓘ principal submatrices ⓘ
describes	relationship between eigenvalues of a symmetric matrix and eigenvalues of its principal submatrices ⓘ
field	linear algebra ⓘ matrix theory ⓘ numerical linear algebra ⓘ spectral theory ⓘ
generalizes	basic eigenvalue interlacing results for leading principal submatrices ⓘ
guarantees	bounds on eigenvalues of principal submatrices ⓘ monotonicity properties of extremal eigenvalues under restriction to subspaces ⓘ
implies	interlacing of eigenvalues of a symmetric matrix and its principal submatrices ⓘ
isPartOf	eigenvalue interlacing theory ⓘ theory of self-adjoint operators ⓘ
namedAfter	Henri Poincaré NERFINISHED ⓘ
relatedTo	Cauchy interlacing theorem NERFINISHED ⓘ Courant–Fischer min–max theorem NERFINISHED ⓘ Rayleigh–Ritz method NERFINISHED ⓘ Weyl inequalities NERFINISHED ⓘ
states	eigenvalues of a principal submatrix lie between eigenvalues of the original matrix in an interlacing pattern ⓘ
topic	eigenvalue inequalities ⓘ principal minors ⓘ spectral separation ⓘ
usedIn	analysis of subspace projection methods ⓘ approximation of eigenvalues ⓘ finite element methods ⓘ perturbation theory of eigenvalues ⓘ spectral graph theory ⓘ

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Cauchy interlacing theorem → relatedTo → Poincaré separation theorem ⓘ