PlanetMath: Frobenius product

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (21)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Frobenius product

(Definition)

If $A = (a_{ij})$ and $B = (b_{ij})$ are real $m\!\times\!n$ matrices, their Frobenius product is defined as

$\displaystyle \langle A,\,B \rangle_F \;:=\; \sum_{i,\,j}a_{ij}b_{ij}.$

It is easily seen that $\langle A,\,B \rangle_F$ is equal to the trace of the matrix $A^\intercal B$ and $AB^\intercal$ , and that the Frobenius product is an inner product of the vector space formed by the $m\!\times\!n$ matrices; it induces the Frobenius norm of this vector space.

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

"Frobenius product" is owned by pahio.

(view preamble | get metadata)

Other names:

Frobenius inner product

Also defines:

Frobenius norm

This object's parent.

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

Cross-references: vector space, inner product, trace, matrices, real
There are 2 references to this entry.

This is version 4 of Frobenius product, born on 2008-07-11, modified 2008-07-12.
Object id is 10769, canonical name is FrobeniusProduct.
Accessed 551 times total.

Classification:

AMS MSC:	15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
	15A60 (Linear and multilinear algebra; matrix theory :: Norms of matrices, numerical range, applications of functional analysis to matrix theory)

Pending Errata and Addenda

None.[ View all 1 ]

Discussion

forum policy

No messages.

Interact