We are in $M_{n\times n}(\mathbb{R})$. Define $=tr(AB^t)$. First I had to prove that this was indeed an inner product and that part is easy.
The second question is, given a fixed matrix $C$, define $f_C(A)=CA-AC$. Find the adjoint of this operator.
I was trying something along the lines of finding $M_C$, the matrix representation of $f_C$, which would be an $n^2\times n^2$ matrix, and then taking the conjugate transpose of this matrix, and obtaining $M^*_C$ which would be the matrix representation for $f_C$, but this route seems a bit too messy.
I also tried playing with the definition of adjoint, that is, the unique linear operator $f^*_C$ such that $
Does anyone know how to tackle it?
Thanks!