Take the derivative of ${\tt tr}\left[X^*P^*PX\right]$ with respect to P, where P is any complex matrix (or linear operator)

Question

I imagine I will have to differentiate with respect to the real and imaginary parts of P separately, but I am honestly having trouble not making sign errors.

Answer 1

Let's use finite differences: $$ X^* \; \left(P + \triangle P\right)^* \; \left(P+ \triangle P \right) \; X = \\ X^* \; \left[ \left(P + \triangle P\right)^* \; \left(P+ \triangle P \right) \right] X = \\ X^{*} \left[ P^* P + P^* \triangle P + P \triangle P^* + O\left( \; \left| \, \triangle P \, \right|^2 \; \right) \right] X $$ Subtract from this the expression $X^* \; P^* \; P \; X$: $$ X^{*} \left[ P^* \triangle P + P \triangle P^* + O\left( \; \left| \, \triangle P \, \right|^2 \; \right) \right] X. $$ Thus, the sought derivative is the linear operator that acts on a transformation $H$ by: $$ H \mapsto {\tt tr} \left( X^{*} \left[ P^* H + P H^* \right] X \right). $$