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 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
and
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
.
Lemma.[1][2] If , and
then the following are equivalent :
(i) .
(ii)
(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.
Notes
- ↑ 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)
- ↑ 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)