## You are here

HomeFrobenius product

## Primary tabs

# Frobenius product

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

$\langle A,\,B\rangle_{F}\;:=\;\sum_{{i,\,j}}a_{{ij}}b_{{ij}}.$ |

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.

Defines:

Frobenius norm

Related:

NormedVectorSpace, FrobeniusMatrixNorm, Product

Synonym:

Frobenius inner product

Type of Math Object:

Definition

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

15A60*no label found*15A63

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections