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.
Theorem. Let and be vectors in such that their entries are in non-increasing order. There is a Hermitian matrix with diagonal values and eigenvalues if and only if
Polyhedral geometry perspective
Permutation polytope generated by a vector
The permutation polytope generated by denoted by is defined as the convex hull of the set . Here denotes the symmetric group on . The following lemma characterizes the permutation polytope of a vector in .
(iii) There are points in such that and for each in , some transposition in , and some in , depending on .
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 and be real vectors. There is a Hermitian matrix with diagonal entries and eigenvalues if and only if the vector is in the permutation polytope generated by .
Note that in this formulation, one does not need to impose any ordering on the entries of the vectors and .
Proof of the Schur–Horn theorem
Let be a Hermitian matrix with eigenvalues , counted with multiplicity. Denote the diagonal of by , thought of as a vector in , and the vector by . Let be the diagonal matrix having on its diagonal.
( ) may be written in the form , where is a unitary matrix. Then
Let be the matrix defined by . Since is a unitary matrix, is a doubly stochastic matrix and we have . By the Birkhoff–von Neumann theorem, can be written as a convex combination of permutation matrices. Thus is in the permutation polytope generated by . This proves Schur's theorem.
( ) If occurs as the diagonal of a Hermitian matrix with eigenvalues , then also occurs as the diagonal of some Hermitian matrix with the same set of eigenvalues, for any transposition in . One may prove that in the following manner.
Let be a complex number of modulus such that and be a unitary matrix with in the and entries, respectively, at the and entries, respectively, at all diagonal entries other than and , and at all other entries. Then has at the entry, at the entry, and at the entry where . Let be the transposition of that interchanges and .
Then the diagonal of is .
is a Hermitian matrix with eigenvalues . Using the equivalence of (i) and (iii) in the lemma mentioned above, we see that any vector in the permutation polytope generated by , 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 denote the group of unitary matrices. Its Lie algebra, denoted by , is the set of skew-Hermitian matrices. One may identify the dual space with the set of Hermitian matrices via the linear isomorphism defined by for . The unitary group acts on by conjugation and acts on by the coadjoint action. Under these actions, is an -equivariant map i.e. for every the following diagram commutes,
Let and denote the diagonal matrix with entries given by . Let denote the orbit of under the -action i.e. conjugation. Under the -equivariant isomorphism , the symplectic structure on the corresponding coadjoint orbit may be brought onto . Thus is a Hamiltonian -manifold.
Let denote the Cartan subgroup of which consists of diagonal complex matrices with diagonal entries of modulus . The Lie algebra of consists of diagonal skew-Hermitian matrices and the dual space consists of diagonal Hermitian matrices, under the isomorphism . In other words, consists of diagonal matrices with purely imaginary entries and consists of diagonal matrices with real entries. The inclusion map induces a map , which projects a matrix to the diagonal matrix with the same diagonal entries as . The set is a Hamiltonian -manifold, and the restriction of to this set is a moment map for this action.
By the Atiyah–Guillemin–Sternberg theorem, is a convex polytope. A matrix is fixed under conjugation by every element of if and only if is diagonal. The only diagonal matrices in are the ones with diagonal entries in some order. Thus, these matrices generate the convex polytope . This is exactly the statement of the Schur–Horn theorem.
- Schur, Issai, Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie, Sitzungsber. Berl. Math. Ges. 22 (1923), 9–20.
- Horn, Alfred, Doubly stochastic matrices and the diagonal of a rotation matrix, American Journal of Mathematics 76 (1954), 620–630.
- Kadison, R. V.; Pedersen, G. K., Means and Convex Combinations of Unitary Operators, Math. Scand. 57 (1985),249–266.
- Kadison, R. V., The Pythagorean Theorem: I. The finite case, Proc. Natl. Acad. Sci. USA, vol. 99 no. 7 (2002):4178–4184 (electronic)
- Terry Tao: 254A, Notes 3a: Eigenvalues and sums of Hermitian matrices
- Sheela Devadas, Peter J. Haine, Keaton Stubis: The Schur-Horn Theorem