# 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 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