Frobenius Inner Product

linear-algebra

Definition

Frobenius Inner Product

Let $A,B\in F^{m\times n}$ be two matrices over a field $F$.

The Frobenius inner product $⟨\cdot ,\cdot {⟩}_F:F^{m\times n}\times F^{m\times n}\to F$ is an inner product defined as

$$⟨A,B{⟩}_F=\sum _{i=1}^m\sum _{j=1}^n\overline{A_{ij}}{B}_{ij}=\mathrm{tr}(A^{\ast }B)$$

For real matrices this simplifies to ${\sum }_{i,j}{A}_{ij}{B}_{ij}$.

Einstein Notation: $⟨A,B{⟩}_F=\overline{A_{ij}}{B}_{ij}$