Characterizes the diagonal of a Hermitian matrix with given eigenvalues
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
![{\displaystyle \sum _{i=1}^{n}d_{i}\leq \sum _{i=1}^{n}\lambda _{i}\qquad n=1,\dots ,N-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/416dde5ad127cad5086c399ad2e6a49d40ae4fdd)
and
![{\displaystyle \sum _{i=1}^{N}d_{i}=\sum _{i=1}^{N}\lambda _{i}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cad451b7155da2d4eb5239e9a77954d18849eb18)
The inequalities above may alternatively be written:
![{\displaystyle {\begin{alignedat}{7}d_{1}&\;\leq \;&&\lambda _{1}\\[0.3ex]d_{2}+d_{1}&\;\leq &&\lambda _{1}+\lambda _{2}\\[0.3ex]\vdots &\;\leq &&\vdots \\[0.3ex]d_{N-1}+\cdots +d_{2}+d_{1}&\;\leq &&\lambda _{1}+\lambda _{2}+\cdots +\lambda _{N-1}\\[0.3ex]d_{N}+d_{N-1}+\cdots +d_{2}+d_{1}&\;=&&\lambda _{1}+\lambda _{2}+\cdots +\lambda _{N-1}+\lambda _{N}.\\[0.3ex]\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc7fbf009a20fb327b6bd55e8000584ecdf86f7f)
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:
![{\displaystyle {\tilde {y}}:=(y_{1},\cdots ,y_{n})\in {\mathcal {K}}_{\tilde {x}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1918b1236da762f21f677f93ba46eef29a3e8e0)
and
and
and ![{\displaystyle y_{1}+y_{2}+\cdots +y_{n-1}\leq x_{1}+x_{2}+\cdots +x_{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1f1d8246aa8e56db5b54f786367a7c50156786)
- 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 ![{\displaystyle k.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcb6778a29f576eb23da1dbddffb73b2571359ac)
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
![{\displaystyle a_{ii}=\sum _{j=1}^{n}\lambda _{j}|u_{ij}|^{2},\;i=1,2,\ldots ,n.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25d5454a5d8096582cf2570a5b57b4e38014442e)
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)
External links
- MathWorld
- Terry Tao: 254A, Notes 3a: Eigenvalues and sums of Hermitian matrices
- Sheela Devadas, Peter J. Haine, Keaton Stubis: The Schur-Horn Theorem
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