Wielandt-Hoffman theorem

Let $A$ and $B$ be normal matrices (http://planetmath.org/NormalMatrix). Let their eigenvalues $a_i$ and $b_i$ be ordered such that ${\sum }_i{|a_i-b_i|}^2$ is minimized. Then we have the following inequality

$$\sum _i{|a_i-b_i|}^2\le {\parallel A-B\parallel }_F^2,$$

where $\parallel \cdot {\parallel }_F$ is the Frobenius matrix norm.

Title	Wielandt-Hoffman theorem
Canonical name	WielandtHoffmanTheorem
Date of creation	2013-03-22 14:58:45
Last modified on	2013-03-22 14:58:45
Owner	Andrea Ambrosio (7332)
Last modified by	Andrea Ambrosio (7332)
Numerical id	4
Author	Andrea Ambrosio (7332)
Entry type	Theorem
Classification	msc 15A42
Classification	msc 15A18
Related topic	ShursInequality