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 $\mathbf{d}=\{d_i\}_{i=1}^N$ and $\mathbf{\lambda}=\{\lambda_i\}_{i=1}^N$ be vectors in $\mathbb {R} ^{N}$ such that their entries are in non-increasing order. There is a Hermitian matrix with diagonal values $\{d_i\}_{i=1}^N$ and eigenvalues $\{\lambda_i\}_{i=1}^N$ if and only if

\sum_{i=1}^n d_i \leq \sum_{i=1}^n \lambda_i \qquad n=1,2,\ldots,N

and

\sum_{i=1}^N d_i= \sum_{i=1}^N \lambda_i.

Polyhedral geometry perspective

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 $\{1, 2, \ldots, n\}$ . The following lemma characterizes the permutation polytope of a vector in $\mathbb {R} ^{n}$ .

Lemma.^[1]^[2] If $x_1 \ge x_2 \ge \cdots \ge x_n, y_1 \ge y_2 \ge \cdots \ge y_n$ , and $x_1 + x_2 + \cdots + x_n = y_1 + y_2 + \cdots + y_n,$ then the following are equivalent :

(i) $(y_1, y_2, \cdots, y_n)(= \tilde{y}) \in \mathcal{K}_{\tilde{x}}$ .

(ii) $y_1 \le x_1, y_1 + y_2 \le x_1 + x_2 , \ldots, y_1 + y_2 + \cdots + y_{n-1} \le x_1 + x_2 + \cdots + x_n$

(iii) There are points $(x_1^{(1)}, x_2 ^{(1)}, \cdots, x_n ^{(1)})(=\tilde{x}_1), \ldots, (x_1^{(n)}, x_2 ^{(n)}, \ldots, x_n^{(n)})(=\tilde{x}_n)$ in $\mathcal{K}_{\tilde{x}}$ such that $\tilde{x}_1=\tilde{x}, \tilde{x}_n=\tilde{y},$ and $\tilde{x}_{k+1} = t \tilde{x}_k + (1-t) \tau(\tilde{x_k})$ 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 $\mathbf{d}=\{d_i\}_{i=1}^N$ and $\mathbf{\lambda}=\{\lambda_i\}_{i=1}^N$ be real vectors. There is a Hermitian matrix with diagonal entries $\{d_i\}_{i=1}^N$ and eigenvalues $\{\lambda_i\}_{i=1}^N$ if and only if the vector $(d_1, d_2, \ldots, d_n)$ is in the permutation polytope generated by $(\lambda_1, \lambda_2, \ldots, \lambda_n)$ .

Note that in this formulation, one does not need to impose any ordering on the entries of the vectors $\mathbf{d}$ and $\mathbf{\lambda}$ .

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, $-\sqrt{1-t^2}, \xi \sqrt{1-t^2}$ 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 $l \ne j,k$ . Let $\tau$ be the transposition of $\{1, 2, \ldots, n\}$ 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.

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

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