Schur–Horn theorem

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

$${\displaystyle \sum _{i=1}^{n}d_{i}\leq \sum _{i=1}^{n}\lambda _{i}\qquad n=1,\dots ,N-1}$$

and

$${\displaystyle \sum _{i=1}^{N}d_{i}=\sum _{i=1}^{N}\lambda _{i}.}$$

The inequalities above may alternatively be written:

$${\displaystyle {\begin{alignedat}{7}d_{1}&\;\leq \;&&\lambda _{1}\\[0.3ex]d_{2}+d_{1}&\;\leq &&\lambda _{1}+\lambda _{2}\\[0.3ex]\vdots &\;\leq &&\vdots \\[0.3ex]d_{N-1}+\cdots +d_{2}+d_{1}&\;\leq &&\lambda _{1}+\lambda _{2}+\cdots +\lambda _{N-1}\\[0.3ex]d_{N}+d_{N-1}+\cdots +d_{2}+d_{1}&\;=&&\lambda _{1}+\lambda _{2}+\cdots +\lambda _{N-1}+\lambda _{N}.\\[0.3ex]\end{alignedat}}}$$

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

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

(${\displaystyle \Rightarrow }$) ${\displaystyle A}$ may be written in the form ${\displaystyle U\Lambda U^{-1},}$ where ${\displaystyle U}$ is a unitary matrix. Then

$${\displaystyle a_{ii}=\sum _{j=1}^{n}\lambda _{j}|u_{ij}|^{2},\;i=1,2,\ldots ,n.}$$

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

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

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

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

${\displaystyle \Lambda }$ is a Hermitian matrix with eigenvalues ${\displaystyle \{\lambda _{i}\}_{i=1}^{n}.}$ Using the equivalence of (i) and (iii) in the lemma mentioned above, we see that any vector in the permutation polytope generated by ${\displaystyle (\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}),}$ occurs as the diagonal of a Hermitian matrix with the prescribed eigenvalues. This proves Horn's theorem.

The Schur–Horn theorem may be viewed as a corollary of the Atiyah–Guillemin–Sternberg convexity theorem in the following manner. Let ${\displaystyle {\mathcal {U}}(n)}$ denote the group of ${\displaystyle n\times n}$ unitary matrices. Its Lie algebra, denoted by ${\displaystyle {\mathfrak {u}}(n),}$ is the set of skew-Hermitian matrices. One may identify the dual space ${\displaystyle {\mathfrak {u}}(n)^{*}}$ with the set of Hermitian matrices ${\displaystyle {\mathcal {H}}(n)}$ via the linear isomorphism ${\displaystyle \Psi$ :{\mathcal {H}}(n)\rightarrow {\mathfrak {u}}(n)^{*}} defined by ${\displaystyle \Psi (A)(B)=\mathrm {tr} (iAB)}$ for ${\displaystyle A\in {\mathcal {H}}(n),B\in {\mathfrak {u}}(n).}$ The unitary group ${\displaystyle {\mathcal {U}}(n)}$ acts on ${\displaystyle {\mathcal {H}}(n)}$ by conjugation and acts on ${\displaystyle {\mathfrak {u}}(n)^{*}}$ by the coadjoint action. Under these actions, ${\displaystyle \Psi }$ is an ${\displaystyle {\mathcal {U}}(n)}$-equivariant map i.e. for every ${\displaystyle U\in {\mathcal {U}}(n)}$ the following diagram commutes,

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

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

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

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• MathWorld
• Terry Tao: 254A, Notes 3a: Eigenvalues and sums of Hermitian matrices
• Sheela Devadas, Peter J. Haine, Keaton Stubis: The Schur-Horn Theorem