Schur–Horn theorem

In mathematics, particularly linear algebra, the Schur–Horn theorem, named after Issai Schur and Alfred Horn, characterizes the diagonal of a Hermitian matrix with given eigenvalues.It has inspired investigations and substantial generalizations in the setting of symplectic geometry.be two sequences of real numbers arranged in a non-increasing order.There is a Hermitian matrix with diagonal valuesThe left hand side of the theorem's characterization (that is, "there exists a Hermitian matrix with these eigenvalues and diagonal elements") depends on the order of the desired diagonal elements(because changing their order would change the Hermitian matrix whose existence is in question) but it does not depend on the order of the desired eigenvaluescompletely unnecessary: The permutation polytope generated byis defined as the convex hull of the setIn other words, the permutation polytope generated byis the convex hull of the set of all points infor instance, is the convex hull of the setwhich in this case is the solid (filled) triangle whose vertices are the three points in this set.does not change the resulting permutation polytope; in other words, if a pointThe following lemma characterizes the permutation polytope of a vector inthen the following statements are equivalent: In view of the equivalence of (i) and (ii) in the lemma mentioned above, one may reformulate the theorem in the following manner.There is a Hermitian matrix with diagonal entriesNote that in this formulation, one does not need to impose any ordering on the entries of the vectorscan be written as a convex combination of permutation matrices.occurs as the diagonal of a Hermitian matrix with eigenvaluesalso occurs as the diagonal of some Hermitian matrix with the same set of eigenvalues, for any transpositionUsing the equivalence of (i) and (iii) in the lemma mentioned above, we see that any vector in the permutation polytope generated byoccurs as the diagonal of a Hermitian matrix with the prescribed eigenvalues.denote the diagonal matrix with entries given bythe symplectic structure on the corresponding coadjoint orbit may be brought ontoconsists of diagonal skew-Hermitian matrices and the dual spaceconsists of diagonal Hermitian matrices, under the isomorphismconsists of diagonal matrices with purely imaginary entries andconsists of diagonal matrices with real entries.to this set is a moment map for this action.Thus, these matrices generate the convex polytope
mathematicslinear algebraIssai SchurAlfred HornHermitian matrixeigenvaluessymplectic geometryKostant's convexity theoremmajorizationsum of the first n {\displaystyle n} symmetric groupconvex hulldoubly stochastic matrixBirkhoff–von Neumann theoremskew-Hermitiancoadjoint actionCartan subgroupmoment mapKadison, R. V.Schur, IssaiHorn, AlfredTerry TaoFunctional analysistopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexReflexiveSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointBoundedCompactHilbert–SchmidtNormalTrace classTransposeUnboundedUnitaryBanach algebraC*-algebraSpectrum of a C*-algebraOperator algebraGroup algebra of a locally compact groupVon Neumann algebraInvariant subspace problemMahler's conjectureHardy spaceSpectral theory of ordinary differential equationsHeat kernelIndex theoremCalculus of variationsFunctional calculusIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theory