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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 $=$ for all $A,B$ but this was not fruitful either.

Does anyone know how to tackle it?

Thanks!

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    $\displaystyle\large \left[C,A\right]^{\dagger} =\left[A^{\dagger},C^{\dagger}\right]$2014-03-17

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