Frobenius product

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

$${⟨A,B⟩}_F:=\sum _{i,j}{a}_{ij}{b}_{ij}.$$

It is easily seen that  ${⟨A,B⟩}_F$  is equal to the trace of the matrix $A^{⊺}B$ and $A{B}^{⊺}$, 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