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Horn's Theorem

Let

X={x_1>=x_2>=...>=x_n|x_i in R}
(1)

and

Y={y_1>=y_2>=...>=y_n|y_i in R}.
(2)

Then there exists an n×n Hermitian matrix with eigenvalues X and diagonal elements Y iff

sum_(i=1)^t(x_i-y_i)>=0
(3)

for all 1<=t<=n and with equality for t=n. The theorem is sometimes also known as Schur's theorem.

See also

Hermitian Matrix, Majorization, Stochastic Matrix

This entry contributed by Fred Manby

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References

Horn, A. "Doubly Stochastic Matrices and the Diagonal of a Rotation Matrix." Amer. J. Math. 76, 620-630, 1954.Lieb, E. H. "Variational Principle for Many-Fermion Systems." Phys. Rev. Lett. 46, 457-459, 1981.

Referenced on Wolfram|Alpha

Horn's Theorem

Cite this as:

Manby, Fred. "Horn's Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HornsTheorem.html

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