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I have a function about the trace of a matrix

$f(\mathbf{h})=\mathrm{tr}[\mathbf{F} (\mathbf{I}_N \otimes \mathrm{diag}(\mathbf{h})) \mathbf{A} (\mathbf{I}_N \otimes \mathrm{diag}(\mathbf{h})) \mathbf{F}^\mathrm{H}]$

where $\mathbf{I}_N$ is the identity matrix.

What is $\frac{\partial f}{\partial \mathbf{h}}$?

The most related question is

Derivative of a trace with Kronecker product

and

Replace $X$ with $\mbox{diag}(x)$ in trace matrix derivative identity

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I will answer my question by myself.

Just a slight modification of the answer in Replace $X$ with $\mbox{diag}(x)$ in trace matrix derivative identity

$\frac{\partial f}{\partial \mathbf{h}} = (\mathbf{1}_{1 \times N} \otimes \mathbf{I}_M) (\mathrm{diag}^{-1}(\mathbf{A}\mathrm{diag}(\mathbf{I}_N \otimes \mathrm{diag}(\mathbf{h}))\mathbf{F}^\mathrm{H}\mathbf{F}) + \mathrm{diag}^{-1}(\mathbf{F}^\mathrm{H}\mathbf{F}\mathrm{diag}(\mathbf{I}_N \otimes \mathrm{diag}(\mathbf{h}))\mathbf{A}))$

where $M$ is the length of $\mathbf{h}$.

But I am still wondering if there is a simpler form.

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