PlanetMath: Schur's inequality

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (511)
Corrections (9)
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

Schur's inequality

(Theorem)

Theorem (Schur's inequality) Let be a square $n\times n$ matrix with real (or possibly complex entries). If $\lambda_1,\ldots, \lambda_n$ are the eigenvalues of , and is the diagonal matrix $D=\operatorname{diag}(\lambda_1,\ldots, \lambda_n)$ , then

$\displaystyle \Vert D \Vert_F$

$\displaystyle \le$

$\displaystyle \Vert A \Vert_F,$

where $\Vert\cdot \Vert_F$ is the Frobenius matrix norm. Equality holds if and only if

is a normal matrix.

References

1: V.V. Prasolov, Problems and Theorems in Linear Algebra, American Mathematical Society, 1994.

Anyone with an account can edit this entry. Please help improve it!

"Schur's inequality" is owned by matte. [ full author list (2) ]

(view preamble)

See Also: trace of a matrix, Wielandt-Hoffman theorem, Frobenius matrix norm

This object's parent.

Attachments:: proof of Schur's inequality (Proof) by Andrea Ambrosio

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

Cross-references: normal matrix, equality, Frobenius matrix norm, diagonal matrix, eigenvalues, complex, real, matrix, square
There are 2 references to this entry.

This is version 11 of Schur's inequality, born on 2003-06-28, modified 2006-06-12.
Object id is 4410, canonical name is ShursInequality.
Accessed 5131 times total.

Classification:

AMS MSC:	15A42 (Linear and multilinear algebra; matrix theory :: Inequalities involving eigenvalues and eigenvectors)
	26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals)

Pending Errata and Addenda

None.[ View all 3 ]

Discussion

forum policy

No messages.

Interact