Weyl's inequality
The treated in this article from Weyl 's inequality is a statement which was Hermann Weyl in 1912. There are several inequalities which are named after Hermann Weyl. The inequality described here makes a statement about the behavior of the sum of eigenvalues of matrices. This theorem was already known in the 19th century, but has not been fully published.
Weyl 's inequality for matrices
Given a square matrix with the dismantling Here, and any square matrices. With that respected the -th eigenvalue understood, with positive among ascending order, and negative to descending. Accordingly, it is the smallest eigenvalue of and the largest. With the shorthand notation, and is the inequality:
For each pair, which satisfies the inequalities hold
And