PlanetMath: Wielandt-Hoffman theorem

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (197)
Orphanage
Unclass'd
Unproven (498)
Corrections (25)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

Wielandt-Hoffman theorem

(Theorem)

Let and be normal matrices. Let their eigenvalues and be ordered such that $\sum_i \vert a_i-b_i\vert^2$ is minimized. Then we have the following inequality

$\displaystyle \sum_i \vert a_i-b_i\vert^2 \le \Vert A-B\Vert _F^2,$

where $\Vert\cdot\Vert _F$ is the Frobenius matrix norm.

"Wielandt-Hoffman theorem" is owned by Andrea Ambrosio. [ owner history (1) ]

(view preamble)

See Also: Schur's inequality

Attachments:: proof of Wielandt-Hoffman theorem (Proof) by Andrea Ambrosio

Log in to rate this entry.
(view current ratings)

Cross-references: Frobenius matrix norm, inequality, eigenvalues
There are 2 references to this entry.

This is version 1 of Wielandt-Hoffman theorem, born on 2005-01-29.
Object id is 6680, canonical name is WielandtHoffmanTheorem.
Accessed 1981 times total.

Classification:

AMS MSC:	15A42 (Linear and multilinear algebra; matrix theory :: Inequalities involving eigenvalues and eigenvectors)
	15A18 (Linear and multilinear algebra; matrix theory :: Eigenvalues, singular values, and eigenvectors)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact