# Frobenius product

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

$${\u27e8A,B\u27e9}_{F}:=\sum _{i,j}{a}_{ij}{b}_{ij}.$$ |

It is easily seen that ${\u27e8A,B\u27e9}_{F}$ is equal to the trace of the matrix ${A}^{\u22ba}B$ and $A{B}^{\u22ba}$, and that the Frobenius product is an inner product of the vector space^{} formed by the $m\times n$ matrices; it the Frobenius norm of this vector space.

Title | Frobenius product |

Canonical name | FrobeniusProduct |

Date of creation | 2013-03-22 18:11:34 |

Last modified on | 2013-03-22 18:11:34 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 15A60 |

Classification | msc 15A63 |

Synonym | Frobenius inner product |

Related topic | NormedVectorSpace |

Related topic | FrobeniusMatrixNorm |

Related topic | Product |

Defines | Frobenius norm |