Weyl's inequality

Weyl's inequality discussed in this article is a statement that Hermann Weyl found in 1912. There are several inequalities that are named after Hermann Weyl. The inequality described here makes a statement about the behavior of eigenvalues ​​of sums of matrices. This sentence was already known in the 19th century, but was not published in full.

Weyl inequality for matrices

A square matrix is ​​given with the decomposition where and are arbitrary square matrices. In each case the -th eigenvalue is understood with, with positive ones belonging to ascending order and negative ones belonging to descending order. It is therefore the smallest eigenvalue of and the largest. With the abbreviations , and the inequality reads: ${\displaystyle A\in \mathbb {R} ^{n\times n}}$ ${\displaystyle A=W+Y.}$ ${\displaystyle W}$ ${\displaystyle Y}$ ${\displaystyle \lambda _{\pm i}\left(X\right)}$ ${\displaystyle i}$ ${\displaystyle i}$ ${\displaystyle i}$ ${\displaystyle \lambda _{1}\left(X\right)}$ ${\displaystyle X}$ ${\displaystyle \lambda _{-1}\left(X\right)}$ ${\displaystyle \alpha _{\pm i}:=\lambda _{\pm i}\left(A\right)}$ ${\displaystyle \omega _{\pm i}:=\lambda _{\pm i}\left(W\right)}$ ${\displaystyle \eta _{\pm i}:=\lambda _{\pm i}\left(Y\right)}$

The inequalities apply to every pair that satisfies ${\displaystyle i,j}$ ${\displaystyle 1\leq i+j-1\leq n}$

${\displaystyle \omega _{i}+\eta _{j}\leq \alpha _{i+j-1}}$

and

${\displaystyle \alpha _{-\left(i+j-1\right)}\leq \omega _{-i}+\eta _{-j}}$.

swell

literature

• Beresford Parlett: The symmetric eigenvalue problem (Classics in applied mathematics; Vol. 20). Society for Industrial and Applied Mathematics (SIAM), Philadelphia 1998, ISBN 0-89871-402-8 (EA Englewood Cliffs 1980)
• Bertram Huppert (Ed.), Hans Schneider (Ed.), Helmut Wielandt (Author): Mathematische Werke, Volume 2: Linear algebra and analysis . Walter de Gruyter, 1996, ISBN 978-3-11-012453-8 .
• Hermann Weyl : The asymptotic distribution law of the eigenvalues ​​of linear partial differential equations (with an application to the theory of cavity radiation) . In: Mathematische Annalen , Vol. 71 (1912), p. 441, ISSN  0025-5831

Individual evidence

1. ^ Helmut Wielandt (author), Bertram Huppert (ed.), Hans Schneider (ed.): Mathematische Werke: Linear algebra and analysis , p. 166.
2. ↑ Beresford Parlett: The symmetric eigenvalue problem , Chapter 10-3, p. 208.