Tukey's test of additivity

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.

Statement

Theorem. Let $\mathbf {d} =\{d_{i}\}_{i=1}^{N}$ and $\mathbf {\lambda } =\{\lambda _{i}\}_{i=1}^{N}$ be real vectors written 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

The above inequalities can be reformulated geometrically by saying that the vector $(d_{1},d_{2},\ldots ,d_{n})$ is in the convex hull of the $n!$ vectors formed by permuting the coordinates of $(\lambda _{1},\lambda _{2},\ldots ,\lambda _{n})$.

References

• Alfred Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, American Journal of Mathematics 76 (1954), 620–630.
• Issai Schur, Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie, Sitzungsber. Berl. Math. Ges. 22 (1923), 9–20.

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External links

• MathWorld