Set of Schur-Horn

In mathematics , the Schur-Horn theorem characterizes the possible eigenvalues ​​of a Hermitian matrix with a given main diagonal .

Formulation of the sentence

A Hermitian matrix with diagonal entries and eigenvalues exists if and only if the Schur-Horn inequalities ${\displaystyle d_{1}\geq d_{2}\geq \ldots \geq d_{n}}$ ${\displaystyle \lambda _{1}\geq \lambda _{2}\geq \ldots \geq \lambda _{n}}$

${\displaystyle d_{1}\leq \lambda _{1}}$

${\displaystyle d_{1}+d_{2}\leq \lambda _{1}+\lambda _{2}}$

${\displaystyle \ldots }$

${\displaystyle d_{1}+\ldots +d_{n-1}\leq \lambda _{1}+\ldots +\lambda _{n-1}}$

and the equation

${\displaystyle d_{1}+\ldots +d_{n}=\lambda _{1}+\ldots +\lambda _{n}}$

are fulfilled.

The necessity of the condition was proved by Issai Schur , and the reverse by Alfred Horn .

literature

• I. Schur: About a class of mean formations with applications to the determinant theory , session report. Berl. Math. Ges. 22 (1923), 9-20.
• A. Horn: Doubly stochastic matrices and the diagonal of a rotation matrix , American Journal of Mathematics 76 (1954), 620-630.
• Andreas Knauf: Mathematical Physics: Classical Mechanics . Springer, 2017, ISBN 9783662557761 , p. 349 ff.

Web links

• Eric W. Weisstein : Horn's Theorem . In: MathWorld (English).
• Terry Tao : 254A, Notes 3a: Eigenvalues ​​and sums of Hermitian matrices
• Sheela Devadas, Peter J. Haine, Keaton Stubis: The Schur-Horn Theorem