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

Schur–Horn theorem  Theorem. Let and be two sequences of real numbers arranged in a non-increasing order. There is a Hermitian matrix with diagonal values (in this order, starting with at the top-left) and eigenvalues if and only if

and

The inequalities above may alternatively be written:

The Schur–Horn theorem may thus be restated more succinctly and in plain English:

Schur–Horn theorem: Given any non-increasing real sequences of desired diagonal elements and desired eigenvalues there exists a Hermitian matrix with these eigenvalues and diagonal elements if and only if these two sequences have the same sum and for every possible integer the sum of the first desired diagonal elements never exceeds the sum of the first desired eigenvalues.

Reformation allowing unordered diagonals and eigenvalues

Although this theorem requires that and be non-increasing, it is possible to reformulate this theorem without these assumptions.

We start with the assumption The 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 eigenvalues

On the right hand right hand side of the characterization, only the values of depend on the assumption Notice that this assumption means that the expression is just notation for the sum of the largest desired eigenvalues. Replacing the expression with this written equivalent makes the assumption completely unnecessary:

Schur–Horn theorem: Given any desired real eigenvalues and a non-increasing real sequence of desired diagonal elements there exists a Hermitian matrix with these eigenvalues and diagonal elements if and only if these two sequences have the same sum and for every possible integer the sum of the first desired diagonal elements never exceeds the sum of the largest desired eigenvalues.

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 In other words, the permutation polytope generated by is the convex hull of the set of all points in that can be obtained by rearranging the coordinates of The permutation polytope of for instance, is the convex hull of the set which in this case is the solid (filled) triangle whose vertices are the three points in this set. Notice, in particular, that rearranging the coordinates of does not change the resulting permutation polytope; in other words, if a point can be obtained from by rearranging its coordinates, then

The following lemma characterizes the permutation polytope of a vector in

Lemma[1][2]  If and have the same sum then the following statements are equivalent:

  1. and and and
  2. There exist a sequence of points in starting with and ending with such that 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 numbers. 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.

Notes

  1. Kadison, R. V., Lemma 5, The Pythagorean Theorem: I. The finite case, Proc. Natl. Acad. Sci. USA, vol. 99 no. 7 (2002):4178–4184 (electronic)
  2. Kadison, R. V.; Pedersen, G. K., Lemma 13, Means and Convex Combinations of Unitary Operators, Math. Scand. 57 (1985),249–266

References

  • 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)
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.