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 and Kirwan convexity theorem.

Statement

Schur–Horn theorem — Theorem. Let $d_{1},\dots ,d_{N}$ and $\lambda _{1},\dots ,\lambda _{N}$ be two sequences of real numbers arranged in a non-increasing order. There is a Hermitian matrix with diagonal values $d_{1},\dots ,d_{N}$ (in this order, starting with $d_{1}$ at the top-left) and eigenvalues $\lambda _{1},\dots ,\lambda _{N}$ if and only if $$\sum _{i=1}^{n}d_{i}\leq \sum _{i=1}^{n}\lambda _{i}\qquad n=1,\dots ,N-1$$ and $$\sum _{i=1}^{N}d_{i}=\sum _{i=1}^{N}\lambda _{i}.$$

The inequalities above may alternatively be written: $${\begin{alignedat}{7}d_{1}&\;\leq \;&&\lambda _{1}\\[0.3ex]d_{2}+d_{1}&\;\leq &&\lambda _{1}+\lambda _{2}\\[0.3ex]\vdots &\;\leq &&\vdots \\[0.3ex]d_{N-1}+\cdots +d_{2}+d_{1}&\;\leq &&\lambda _{1}+\lambda _{2}+\cdots +\lambda _{N-1}\\[0.3ex]d_{N}+d_{N-1}+\cdots +d_{2}+d_{1}&\;=&&\lambda _{1}+\lambda _{2}+\cdots +\lambda _{N-1}+\lambda _{N}.\\[0.3ex]\end{alignedat}}$$

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 $d_{1}\geq \cdots \geq d_{N}$ and desired eigenvalues $\lambda _{1}\geq \cdots \geq \lambda _{N},$ 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 $n,$ the sum of the first $n$ desired diagonal elements never exceeds the sum of the first $n$ desired eigenvalues.

Reformation allowing unordered diagonals and eigenvalues

Although this theorem requires that $d_{1}\geq \cdots \geq d_{N}$ and $\lambda _{1}\geq \cdots \geq \lambda _{N}$ be non-increasing, it is possible to reformulate this theorem without these assumptions.

We start with the assumption $\lambda _{1}\geq \cdots \geq \lambda _{N}.$ 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 $d_{1},\dots ,d_{N}$ (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 $\lambda _{1},\dots ,\lambda _{N}.$

On the right hand right hand side of the characterization, only the values of $\lambda _{1}+\cdots +\lambda _{n}$ depend on the assumption $\lambda _{1}\geq \cdots \geq \lambda _{N}.$ Notice that this assumption means that the expression $\lambda _{1}+\cdots +\lambda _{n}$ is just notation for the sum of the $n$ largest desired eigenvalues. Replacing the expression $\lambda _{1}+\cdots +\lambda _{n}$ with this written equivalent makes the assumption $\lambda _{1}\geq \cdots \geq \lambda _{N}$ completely unnecessary:

Schur–Horn theorem: Given any $N$ desired real eigenvalues and a non-increasing real sequence of desired diagonal elements $d_{1}\geq \cdots \geq d_{N},$ 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 $n,$ the sum of the first $n$ desired diagonal elements never exceeds the sum of the $n$ largest desired eigenvalues.

Permutation polytope generated by a vector

The permutation polytope generated by ${\tilde {x}}=(x_{1},x_{2},\ldots ,x_{n})\in \mathbb {R} ^{n}$ denoted by ${\mathcal {K}}_{\tilde {x}}$ is defined as the convex hull of the set $\{(x_{\pi (1)},x_{\pi (2)},\ldots ,x_{\pi (n)})\in \mathbb {R} ^{n}:\pi \in S_{n}\}.$ Here $S_{n}$ denotes the symmetric group on $\{1,2,\ldots ,n\}.$ In other words, the permutation polytope generated by $(x_{1},\dots ,x_{n})$ is the convex hull of the set of all points in $\mathbb {R} ^{n}$ that can be obtained by rearranging the coordinates of $(x_{1},\dots ,x_{n}).$ The permutation polytope of $(1,1,2),$ for instance, is the convex hull of the set $\{(1,1,2),(1,2,1),(2,1,1)\},$ 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 $(x_{1},\dots ,x_{n})$ does not change the resulting permutation polytope; in other words, if a point ${\tilde {y}}$ can be obtained from ${\tilde {x}}=(x_{1},\dots ,x_{n})$ by rearranging its coordinates, then ${\mathcal {K}}_{\tilde {y}}={\mathcal {K}}_{\tilde {x}}.$

The following lemma characterizes the permutation polytope of a vector in $\mathbb {R} ^{n}.$

Lemma — If $x_{1}\geq \cdots \geq x_{n},$ and $y_{1}\geq \cdots \geq y_{n},$ have the same sum $x_{1}+\cdots +x_{n}=y_{1}+\cdots +y_{n},$ then the following statements are equivalent:

1. ${\tilde {y}}:=(y_{1},\cdots ,y_{n})\in {\mathcal {K}}_{\tilde {x}}.$
2. $y_{1}\leq x_{1},$ and $y_{1}+y_{2}\leq x_{1}+x_{2},$ and $\ldots ,$ and $y_{1}+y_{2}+\cdots +y_{n-1}\leq x_{1}+x_{2}+\cdots +x_{n-1}$
3. There exist a sequence of points ${\tilde {x}}_{1},\dots ,{\tilde {x}}_{n}$ in ${\mathcal {K}}_{\tilde {x}},$ starting with ${\tilde {x}}_{1}={\tilde {x}}$ and ending with ${\tilde {x}}_{n}={\tilde {y}}$ such that ${\tilde {x}}_{k+1}=t{\tilde {x}}_{k}+(1-t)\tau ({\tilde {x_{k}}})$ for each $k$ in $\{1,2,\ldots ,n-1\},$ some transposition $\tau$ in $S_{n},$ and some $t$ in $[0,1],$ depending on $k.$

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 $d_{1},\dots ,d_{N}$ and $\lambda _{1},\dots ,\lambda _{N}$ be real numbers. There is a Hermitian matrix with diagonal entries $d_{1},\dots ,d_{N}$ and eigenvalues $\lambda _{1},\dots ,\lambda _{N}$ if and only if the vector $(d_{1},\ldots ,d_{n})$ is in the permutation polytope generated by $(\lambda _{1},\ldots ,\lambda _{n}).$

Note that in this formulation, one does not need to impose any ordering on the entries of the vectors $d_{1},\dots ,d_{N}$ and $\lambda _{1},\dots ,\lambda _{N}.$

Proof of the Schur–Horn theorem

Let $A=(a_{jk})$ be a $n\times n$ Hermitian matrix with eigenvalues $\{\lambda _{i}\}_{i=1}^{n},$ counted with multiplicity. Denote the diagonal of $A$ by ${\tilde {a}},$ thought of as a vector in $\mathbb {R} ^{n},$ and the vector $(\lambda _{1},\lambda _{2},\ldots ,\lambda _{n})$ by ${\tilde {\lambda }}.$ Let $\Lambda$ be the diagonal matrix having $\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}$ on its diagonal.

($\Rightarrow$) $A$ may be written in the form $U\Lambda U^{-1},$ where $U$ is a unitary matrix. Then $$a_{ii}=\sum _{j=1}^{n}\lambda _{j}|u_{ij}|^{2},\;i=1,2,\ldots ,n.$$

Let $S=(s_{ij})$ be the matrix defined by $s_{ij}=|u_{ij}|^{2}.$ Since $U$ is a unitary matrix, $S$ is a doubly stochastic matrix and we have ${\tilde {a}}=S{\tilde {\lambda }}.$ By the Birkhoff–von Neumann theorem, $S$ can be written as a convex combination of permutation matrices. Thus ${\tilde {a}}$ is in the permutation polytope generated by ${\tilde {\lambda }}.$ This proves Schur's theorem.

($\Leftarrow$) If ${\tilde {a}}$ occurs as the diagonal of a Hermitian matrix with eigenvalues $\{\lambda _{i}\}_{i=1}^{n},$ then $t{\tilde {a}}+(1-t)\tau ({\tilde {a}})$ also occurs as the diagonal of some Hermitian matrix with the same set of eigenvalues, for any transposition $\tau$ in $S_{n}.$ One may prove that in the following manner.

Let $\xi$ be a complex number of modulus $1$ such that ${\overline {\xi a_{jk}}}=-\xi a_{jk}$ and $U$ be a unitary matrix with $\xi {\sqrt {t}},{\sqrt {t}}$ in the $j,j$ and $k,k$ entries, respectively, $-{\sqrt {1-t}},\xi {\sqrt {1-t}}$ at the $j,k$ and $k,j$ entries, respectively, $1$ at all diagonal entries other than $j,j$ and $k,k,$ and $0$ at all other entries. Then $UAU^{-1}$ has $ta_{jj}+(1-t)a_{kk}$ at the $j,j$ entry, $(1-t)a_{jj}+ta_{kk}$ at the $k,k$ entry, and $a_{ll}$ at the $l,l$ entry where $l\neq j,k.$ Let $\tau$ be the transposition of $\{1,2,\dots ,n\}$ that interchanges $j$ and $k.$

Then the diagonal of $UAU^{-1}$ is $t{\tilde {a}}+(1-t)\tau ({\tilde {a}}).$

$\Lambda$ is a Hermitian matrix with eigenvalues $\{\lambda _{i}\}_{i=1}^{n}.$ Using the equivalence of (i) and (iii) in the lemma mentioned above, we see that any vector in the permutation polytope generated by $(\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}),$ 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 ${\mathcal {U}}(n)$ denote the group of $n\times n$ unitary matrices. Its Lie algebra, denoted by ${\mathfrak {u}}(n),$ is the set of skew-Hermitian matrices. One may identify the dual space ${\mathfrak {u}}(n)^{*}$ with the set of Hermitian matrices ${\mathcal {H}}(n)$ via the linear isomorphism $\Psi :{\mathcal {H}}(n)\rightarrow {\mathfrak {u}}(n)^{*}$ defined by $\Psi (A)(B)=\mathrm {tr} (iAB)$ for $A\in {\mathcal {H}}(n),B\in {\mathfrak {u}}(n).$ The unitary group ${\mathcal {U}}(n)$ acts on ${\mathcal {H}}(n)$ by conjugation and acts on ${\mathfrak {u}}(n)^{*}$ by the coadjoint action. Under these actions, $\Psi$ is an ${\mathcal {U}}(n)$-equivariant map i.e. for every $U\in {\mathcal {U}}(n)$ the following diagram commutes,

Let ${\tilde {\lambda }}=(\lambda _{1},\lambda _{2},\ldots ,\lambda _{n})\in \mathbb {R} ^{n}$ and $\Lambda \in {\mathcal {H}}(n)$ denote the diagonal matrix with entries given by ${\tilde {\lambda }}.$ Let ${\mathcal {O}}_{\tilde {\lambda }}$ denote the orbit of $\Lambda$ under the ${\mathcal {U}}(n)$-action i.e. conjugation. Under the ${\mathcal {U}}(n)$-equivariant isomorphism $\Psi ,$ the symplectic structure on the corresponding coadjoint orbit may be brought onto ${\mathcal {O}}_{\tilde {\lambda }}.$ Thus ${\mathcal {O}}_{\tilde {\lambda }}$ is a Hamiltonian ${\mathcal {U}}(n)$-manifold.

Let $\mathbb {T}$ denote the Cartan subgroup of ${\mathcal {U}}(n)$ which consists of diagonal complex matrices with diagonal entries of modulus $1.$ The Lie algebra ${\mathfrak {t}}$ of $\mathbb {T}$ consists of diagonal skew-Hermitian matrices and the dual space ${\mathfrak {t}}^{*}$ consists of diagonal Hermitian matrices, under the isomorphism $\Psi .$ In other words, ${\mathfrak {t}}$ consists of diagonal matrices with purely imaginary entries and ${\mathfrak {t}}^{*}$ consists of diagonal matrices with real entries. The inclusion map ${\mathfrak {t}}\hookrightarrow {\mathfrak {u}}(n)$ induces a map $\Phi :{\mathcal {H}}(n)\cong {\mathfrak {u}}(n)^{*}\rightarrow {\mathfrak {t}}^{*},$ which projects a matrix $A$ to the diagonal matrix with the same diagonal entries as $A.$ The set ${\mathcal {O}}_{\tilde {\lambda }}$ is a Hamiltonian $\mathbb {T}$-manifold, and the restriction of $\Phi$ to this set is a moment map for this action.

By the Atiyah–Guillemin–Sternberg theorem, $\Phi ({\mathcal {O}}_{\tilde {\lambda }})$ is a convex polytope. A matrix $A\in {\mathcal {H}}(n)$ is fixed under conjugation by every element of $\mathbb {T}$ if and only if $A$ is diagonal. The only diagonal matrices in ${\mathcal {O}}_{\tilde {\lambda }}$ are the ones with diagonal entries $\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}$ in some order. Thus, these matrices generate the convex polytope $\Phi ({\mathcal {O}}_{\tilde {\lambda }}).$ This is exactly the statement of the Schur–Horn theorem.

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