Let $x$ and $y\in \mathbb{C}^{K\times 1}$ and $H\in \mathbb{C}^{K\times K}$ a diagonal matrix.
$\bar{x}$ denotes the complex conjugated, $x^{T}$ denotes the transpose and $x^{*}$ denotes the complex conjuguate transpose $x^{*}=\bar{x}^{T}$.
What are :
- $\frac{\partial }{\partial x}x^{*}Hx$ and $\frac{\partial }{\partial x}x^{T}Hx$ ?
- $\frac{\partial }{\partial x}x^{*}Hy$ and $\frac{\partial }{\partial x}x^{T}Hy$ ?
- $\frac{\partial }{\partial x}x^{*}H\bar{x}$ and $\frac{\partial }{\partial x}x^{T}H\bar{x}$ ?
- $\frac{\partial }{\partial x}x^{*}H\bar{y}$ and $\frac{\partial }{\partial x}x^{T}H\bar{y}$ ?
Best Answer
Denote the transpose, complex conjugate, and hermitian conjugates of $X$ as $(\,X^T,\,X^C,\,X^H),\,$ respectively.
Further, a colon will denote the trace/Frobenius product, i.e. $\,\,A\!:\!B={\rm tr}(A^TB)$
Now consider a single scalar function and its differential $$\eqalign{ \phi &= z^THw = H:zw^T \cr d\phi &= H:(dz\,w^T + z\,dw^T) \cr &= Hw:dz + z^TH:dw^T \cr &= Hw:dz + H^Tz:dw \cr }$$ But since $$ d\phi = \bigg(\frac{\partial\phi}{\partial z}\bigg)\,dz + \bigg(\frac{\partial\phi}{\partial w}\bigg)\,dw $$ we can identify $$\eqalign{ &\frac{\partial\phi}{\partial z} = Hw,\,\,\,\,\,&\frac{\partial\phi}{\partial w} = H^Tz \cr &\frac{\partial\phi}{\partial z^C} = 0,\,\,\,\,\,&\frac{\partial\phi}{\partial w^C} = 0 \cr }$$ Now substitute $\,z = \{x,x^C\}\,$ and $\,w = \{y,y^C,x,x^C\}\,$ to answer each of your questions.