Standard Inner Product for Matrix Spaces

Definition

• With $A:=(a_{j,k})\in {\mathbb{C}}^{n\times n}$
• The $A^{\ast }:=(\overline{a_{k,j}})$ being the Conjugate Transpose of $A$
• We define $⟨A|B⟩=tr(A{B}^{\ast })={\sum }_{i=1}^n(A{B}^{\ast }{)}_{ii}$