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. A few important generalizations are Kostant's convexity theorem, Atiyah–Guillemin–Sternberg convexity theorem, Kirwan convexity theorem. Statement. Theorem. Let formula_1 and formula_2 be two sequences of real numbers arranged in a non-increasing order. There is a Hermitian matrix with diagonal values formula_3 and eigenvalues formula_4 if and only if and Polyhedral geometry perspective. Permutation polytope generated  by a vector. The permutation polytope generated by formula_7 denoted by formula_8 is defined as the convex hull of the set formula_9. Here formula_10 denotes the symmetric group on formula_11. The following lemma characterizes the permutation polytope of a vector in formula_12. </math>.</li> </ol> Reformulation of Schur–Horn theorem. In view of the equivalence of (i) and (ii) in the lemma mentioned above, one may reformulate the theorem in the following manner. Theorem. Let formula_1 and formula_2 be real vectors. There is a Hermitian matrix with diagonal entries formula_3 and eigenvalues formula_4 if and only if  the vector formula_17 is in the permutation polytope generated by formula_18. Note that in this formulation, one does not need to impose any ordering on the entries of the vectors formula_19 and formula_20. Proof of the Schur–Horn theorem. Let formula_21 be a formula_22 Hermitian matrix with eigenvalues formula_23, counted with multiplicity. Denote the diagonal of formula_24 by formula_25, thought of as a vector in formula_12, and the vector formula_18 by formula_28. Let formula_29 be the diagonal matrix having formula_30 on its diagonal. (formula_31) formula_24 may be written in the form formula_33, where formula_34 is a unitary matrix. Then Let  formula_36 be the matrix defined by formula_37. Since formula_34 is a unitary matrix, formula_39 is a doubly stochastic matrix and we have formula_40. By the Birkhoff–von Neumann theorem, formula_39 can be written as a convex combination of permutation matrices. Thus formula_25 is in the permutation polytope generated by formula_28. This proves Schur's theorem. (formula_44) If formula_25 occurs as the diagonal of a Hermitian matrix with eigenvalues formula_46, then formula_47 also occurs as the diagonal of some Hermitian matrix with the same set of eigenvalues, for any transposition formula_48 in formula_10. One may prove that in the following manner . Let formula_50 be a complex number of modulus formula_51 such that formula_52 and formula_34 be a unitary matrix with formula_54 in the formula_55 and formula_56 entries, respectively, formula_57 at the formula_58 and formula_59 entries, respectively, formula_51 at all diagonal entries other than formula_55 and formula_56, and formula_63 at all other entries. Then formula_64 has formula_65 at the formula_55 entry, formula_67 at the formula_56 entry, and formula_69 at the formula_70 entry where formula_71. Let formula_48 be the transposition of formula_11 that interchanges formula_74 and formula_75. Then the diagonal of formula_64 is formula_47. formula_29 is a Hermitian matrix with eigenvalues formula_46. Using  the equivalence of (i) and (iii) in the lemma mentioned above, we see that any vector in the permutation polytope generated by formula_18, occurs as the diagonal of a Hermitian matrix with the prescribed eigenvalues. This proves Horn's theorem. Symplectic geometry perspective. The Schur–Horn theorem may be viewed as a corollary of the Atiyah–Guillemin–Sternberg convexity theorem in the following manner. Let formula_81 denote the group of formula_22 unitary matrices. Its Lie algebra, denoted by formula_83, is the set of skew-Hermitian matrices. One may identify the dual space formula_84 with the set of Hermitian matrices formula_85  via the linear isomorphism formula_86 defined by formula_87 for formula_88. The unitary group formula_81 acts on formula_85 by conjugation and acts on formula_84 by the coadjoint action. Under these actions, formula_92 is an formula_81-equivariant map i.e. for every formula_94 the following diagram commutes, Let formula_95 and formula_96 denote the diagonal matrix with entries given by formula_28. Let formula_98 denote the orbit of formula_29 under the formula_81-action i.e. conjugation. Under the formula_81-equivariant isomorphism formula_92, the symplectic structure on the corresponding coadjoint orbit may be brought onto formula_98. Thus formula_98 is a Hamiltonian formula_81-manifold. Let formula_106 denote the  Cartan subgroup of formula_81 which consists of diagonal complex matrices with diagonal entries of modulus formula_51. The Lie algebra formula_109 of formula_106 consists of diagonal skew-Hermitian matrices and the dual space formula_111 consists of diagonal Hermitian matrices, under the isomorphism formula_92. In other words, formula_109 consists of diagonal matrices with purely imaginary entries and formula_111 consists of diagonal matrices with real entries. The inclusion map formula_115 induces a map formula_116, which projects a matrix formula_24 to the diagonal matrix with the same diagonal entries as formula_24. The set formula_98 is a Hamiltonian formula_106-manifold, and the restriction of formula_121  to this set is a moment map for this action. By the Atiyah–Guillemin–Sternberg theorem, formula_122 is a convex polytope. A matrix formula_123 is fixed under conjugation by every element of formula_106 if and only if formula_24 is diagonal. The only diagonal matrices in formula_98 are the ones with diagonal entries formula_30 in some order. Thus, these matrices generate the convex polytope formula_122. This is exactly the statement of the Schur–Horn theorem.