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.

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

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

Notice that this assumption means that the expression

completely unnecessary: The permutation polytope generated by

is defined as the convex hull of the set

In other words, the permutation polytope generated by

is the convex hull of the set of all points in

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.

does not change the resulting permutation polytope; in other words, if a point

The following lemma characterizes the permutation polytope of a vector in

then 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 entries

Note that in this formulation, one does not need to impose any ordering on the entries of the vectors

can be written as a convex combination of permutation matrices.

occurs as the diagonal of a Hermitian matrix with eigenvalues

also occurs as the diagonal of some Hermitian matrix with the same set of eigenvalues, for any transposition

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.

the symplectic structure on the corresponding coadjoint orbit may be brought onto

consists of diagonal skew-Hermitian matrices and the dual space

consists of diagonal Hermitian matrices, under the isomorphism

consists of diagonal matrices with purely imaginary entries and

consists of diagonal matrices with real entries.

to this set is a moment map for this action.

Thus, these matrices generate the convex polytope