The Adjoint of Matrix Multiplication
\[V\]
be the vector space of \[n \times n\]
matrices over a field \[K\]
with the inner product \[\lt A, \; B \gt =Tr(B^*A)\]
.The adjoint of left multiplication by
\[M\]
is left multiplication by \[M^*\]
but left multiplication by \[M^*\]
is the linear operator \[L_{M^*}\]
such that \[L_{M^*}(A)=M^*A\]
.\[\begin{equation} \begin{aligned} \lt L_M(A), \; B \gt &= Tr(B^*(MA)) \\ &= Tr(MAB^*) \\ &= Tr(AB^*M) \\ &=Tr(A(M^*B)^*) \\ &= \lt A, L_(M^*(B) \gt \end{aligned} \end{equation}\]
Hence
\[(L_{M})^*=L_{M^*}\]
. 
