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 $\mathit{\lambda }=\{{\lambda }_i{\}}_{i=1}^N$ 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

$\sum _{i=1}^n{d}_i\le \sum _{i=1}^n{\lambda }_{i}n=1,2,\dots ,N$

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 $(d_1,d_2,\dots ,d_n)$ is in the convex hull of the $n!$ vectors formed by permuting the coordinates of $({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n)$.

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