Consider $\mathbf{s}[n]\in\mathbb{R}^{C}, n=1,...,N$ and $\mathbf{S}[f]=\mathcal{F}(\mathbf{s})\in\mathbb{C}^{C}$.
I have a set of matrices $\mathbf{J}[f]=\mathbf{A}\circ\exp\left(-\frac{2\pi \jmath f \mathbf{T}}{N}\right) \in \mathbb{C}^{C\times C}$ with $\mathbf{A},\mathbf{T}\in\mathbb{R}^{C\times C}$ (that are independent of $f$). Please ote that the exponential here is elementwise, i.e. $[\exp(\mathbf{X})]_{ij}=e^{X_{ij}}$.
Where $\tilde{\mathbf{S}}[f]:=\mathbf{J}[f]\mathbf{S}[f]$, I am trying to solve for $$ \min_{\mathbf{J}} \sum_f \|\tilde{\mathbf{S}}[f]- \mathbf{Y}[f]\|^2 $$ where $\mathbf{Y}[f]\in\mathbb{C}^{C}=\mathcal{F}(\mathbf{y})$ is just another matrix.
Differentiating w.r.t. $\mathbf{A}$ and $\mathbf{T}$ led me to \begin{align} \newcommand{\de}{\partial} \frac{\de}{\de \mathbf{A}} \sum_f\|\tilde{\mathbf{S}}[f]-\mathbf{Y}[f]\|^2 &= 2\sum_f\Re \left[\left(\left(\tilde{\mathbf{S}}[f]-\mathbf{Y}[f]\right)\mathbf{S}^H[f]\right) \circ \exp\left(\frac{2\pi\jmath f}{N} \mathbf{T}\right)\right] \end{align}
and
\begin{align} \frac{\de}{\de \mathbf{T}} \sum_f\|\tilde{\mathbf{S}}[f]-\mathbf{Y}\|^2 &= 2\sum_f\Re\left[\left(\frac{2\pi\jmath f}{N} \right)\exp\left(\frac{2\pi\jmath f}{N} \mathbf{T}\right) \circ \mathbf{A} \circ \left(\tilde{\mathbf{S}}[f]-\mathbf{Y}[f]\right)\mathbf{S}^H[f]\right] \end{align}
(I hope that the derivatives are calculated correctly, and would really appreciate it if you could verify that, but it's not the main point here.)
This is where I am stuck, as I could not extract $\mathbf{A}$ and $\mathbf{T}$ out of the joint conditions.
So I did a bit of a "hack": for both derivatives to be zero, one way is to solve for $\left(\tilde{\mathbf{S}}[f]-\mathbf{Y}[f]\right)\mathbf{S}^H[f]=\mathbf{0}$ for all $f$.
This gave me $$\mathbf{J}[f] = \left(\mathbf{S}[f]\mathbf{S}^H[f]\right)^{-1}\left(\mathbf{Y}[f]\mathbf{S}^H[f]\right) \qquad \forall f$$.
But I am stuck again unable to extract the $f$-independent $\mathbf{A}$ and $\mathbf{T}$ from this $f$-dependent solutions of $\mathbf{J}[f]$, or to even verify that this solution takes the form of $\mathbf{J}[f]=\mathbf{A}\circ\exp\left(-\frac{2\pi \jmath f \mathbf{T}}{N}\right)$ as per the definition.
I would really appreciate any help/suggestions to continue with this, or an alternative approach. Thank you!