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 and Kirwan convexity theorem.
Statement
The condition on the two sequences is equivalent to the majorization condition: d → ⪯ λ → {\displaystyle {\vec {d}}\preceq {\vec {\lambda }}} . The inequalities above may alternatively be written:
d 1 ≤ λ 1 d 2 + d 1 ≤ λ 1 + λ 2 ⋮ ≤ ⋮ d N − 1 + ⋯ + d 2 + d 1 ≤ λ 1 + λ 2 + ⋯ + λ N − 1 d N + d N − 1 + ⋯ + d 2 + d 1 = λ 1 + λ 2 + ⋯ + λ N − 1 + λ N . {\displaystyle {\begin{alignedat}{7}d_{1}&\;\leq \;&&\lambda _{1}\\[0.3ex]d_{2}+d_{1}&\;\leq &&\lambda _{1}+\lambda _{2}\\[0.3ex]\vdots &\;\leq &&\vdots \\[0.3ex]d_{N-1}+\cdots +d_{2}+d_{1}&\;\leq &&\lambda _{1}+\lambda _{2}+\cdots +\lambda _{N-1}\\[0.3ex]d_{N}+d_{N-1}+\cdots +d_{2}+d_{1}&\;=&&\lambda _{1}+\lambda _{2}+\cdots +\lambda _{N-1}+\lambda _{N}.\\[0.3ex]\end{alignedat}}}
The Schur–Horn theorem may thus be restated more succinctly and in plain English:
Schur–Horn theorem: Given any non-increasing real sequences of desired diagonal elements d 1 ≥ ⋯ ≥ d N {\displaystyle d_{1}\geq \cdots \geq d_{N}} and desired eigenvalues λ 1 ≥ ⋯ ≥ λ N , {\displaystyle \lambda _{1}\geq \cdots \geq \lambda _{N},} there exists a Hermitian matrix with these eigenvalues and diagonal elements if and only if these two sequences have the same sum and for every possible integer n , {\displaystyle n,} the sum of the first n {\displaystyle n} desired diagonal elements never exceeds the sum of the first n {\displaystyle n} desired eigenvalues.
… excerpt ends here. Continue reading the full article.
