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Consider the following function:

$$\Omega \left( \mathbf{F},\mathbf{E} \right)={{\left( \text{diag}\left( \mathbf{F} \right)\mathbf{M\sigma }-\mathbf{XE} \right)}^{\text{T}}}\mathbf{W}\left( \text{diag}\left( \mathbf{F} \right)\mathbf{M\sigma }-\mathbf{XE} \right),$$

where $\mathbf{E}$and $\mathbf{F}$ are vectors of size $n$ and $m$, respectively, $\mathbf{M}$ is an $m×k$ matrix with a single 1 in its each row (all the other entries are zero), $\mathbf{\sigma }$ is a vector of size $k$, $\mathbf{X}$ is a matrix of size $k×n$, $\mathbf{W}$ is a diagonal matrix of size $k×k$ and $\text{diag}\left( \mathbf{F} \right)$ is a diagonal matrix constructed from the entries of $\mathbf{F}$. Could you please help me how to obtain $\left( \mathbf{\hat{F}},\mathbf{\hat{E}} \right)$, for which $\Omega \left( \mathbf{\hat{F}},\mathbf{\hat{E}} \right)$ is the minimum of $\Omega \left( \mathbf{F},\mathbf{E} \right)$. Using the first derivatives of $\Omega \left( \mathbf{F},\mathbf{E} \right)$, we can derive equations

$$\mathbf{\hat{E}}={{\left( {{\mathbf{X}}^{\text{T}}}\mathbf{WX} \right)}^{-1}}{{\mathbf{X}}^{\text{T}}}\mathbf{W}\text{diag}\left( {\mathbf{\hat{F}}} \right)\mathbf{M\sigma }$$

and

$$\begin{align} & {{\left( \frac{\partial \text{diag}\left( \mathbf{F} \right)}{\partial \mathbf{F}}\mathbf{M\sigma } \right)}^{\text{T}}}\mathbf{W}\text{diag}\left( \mathbf{\hat{F}} \right)\mathbf{M\sigma }= \\ & {{\left( \frac{\partial \text{diag}\left( \mathbf{F} \right)}{\partial \mathbf{F}}\mathbf{M\sigma } \right)}^{\text{T}}}\mathbf{WX}{{\left( {{\mathbf{X}}^{\text{T}}}\mathbf{WX} \right)}^{-1}}{{\mathbf{X}}^{\text{T}}}\mathbf{W}\text{diag}\left( {\mathbf{\hat{F}}} \right)\mathbf{M\sigma } \\ \end{align}$$

from ${{\left( {\partial \Omega \left( \mathbf{F},\mathbf{E} \right)}/{\partial \mathbf{E}}\; \right)}_{\left( \mathbf{\hat{F}},\mathbf{\hat{E}} \right)}}=\mathbf{0}$ and ${{\left( {\partial \Omega \left( \mathbf{F},\mathbf{E} \right)}/{\partial \mathbf{F}}\; \right)}_{\left( \mathbf{\hat{F}},\mathbf{\hat{E}} \right)}}=\mathbf{0}$, respectively. Is there any idea to express $\mathbf{\hat{F}}$ from the latter equation?

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