$\frac{\partial}{\partial X^H} Tr ( X X^H X X ^H)$

Question

Hi I know how to derive the following result below, \begin{eqnarray} \frac{\partial}{\partial X} Tr (X^\top A X B)= A^\top X B^\top +AXB\\ \frac{\partial}{\partial X^\top} Tr (X^\top A X B)=\left( A^\top X B^\top +AXB\right)^\top \end{eqnarray} where $X^\top$ is the real transpose of $X$. However I am trying to derive the following
\begin{eqnarray} \frac{\partial}{\partial X^H} Tr ( X X^H X X ^H)=? \end{eqnarray} where $X^H$ is the complex (hermitian) transpose of the matrix $X$, and the real transpose is given by $X^\top$. How can we approach this? Thanks!

Answer 1

Consider the function $$\eqalign{ f(Y) &= \operatorname{tr}(AYAY) = A^T:YAY \cr }$$where colon denotes the Frobenius (aka double-dot) product.

Let's find the differential and gradient of this function $$\eqalign{ df &= A^T:(dY\,AY+YA\,dY) \cr &= 2\,A^TY^TA^T:dY \cr \cr \frac{\partial f}{\partial Y} &= 2\,A^TY^TA^T \cr \cr }$$ To apply this to your question, set $\,Y\!=\!X^H\,\,$ and $\,\,A\!=\!X,\,\,$ yielding $$\eqalign{ 2\,X^TX^*X^T \cr }$$

Update
(To address some of the questions in the comments)

Since $$\eqalign{ \operatorname{tr}(AYAY) &= \operatorname{tr}(YAYA) \cr }$$the differential and gradient wrt $A$ can be written down by interchanging $Y\leftrightarrow A$ in the previous result, i.e. $$\eqalign{ df &= 2\,Y^TA^TY^T:dA \cr \cr \frac{\partial f}{\partial A} &= 2\,Y^TA^TY^T \cr \cr }$$ If you consider the function $f(A,Y)$, then its full differential is $$\eqalign{ df &= 2\,Y^TA^TY^T:dA \,\,+\,\, 2\,A^TY^TA^T:dY \cr\cr }$$ Rearrangements of the Frobenius product follow from its equivalence to the trace $$A:BC=\operatorname{tr}(A^TBC)$$ and the properties of the trace wrt transposing and/or cyclically shuffling its arguments.

So, for example, the following are all equal $$\eqalign{ A:BC &= AC^T:B \cr &= B^TA:C \cr &= A^T:(BC)^T \cr }$$

Answer 2

Write it down in terms of components, you get $$\frac{\partial}{\partial \bar{X}_{ij}} \mathop{\rm tr}(X X^H X X^H) =\frac{\partial}{\partial \bar{X}_{ij}} ( X_{kl} \bar{X}_{ml} X_{mn} \bar{X}_{kn}), $$ where for some components you need to compute $\frac{d\bar{z}}{dz}$ which doesn't exist, since the function $z\mapsto \bar{z}$ is not differentiable (said not holomorphic.)