矩阵求导公式、正交傅里叶矩阵变换

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矩阵求导公式
矩阵求导：
$\frac{\partial (ax+b)^2}{\partial x}=a^T(ax+b)$
矩阵求导、几种重要的矩阵及常用的矩阵求导公
https://blog.csdn.net/daaikuaichuan/article/details/80620518

$\widehat{\mathbf{a}}=\sqrt{T} \mathbf{F} \mathbf{a}$,

matlab验证：

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$\arg \min _{\mathbf{h}_{k}} \left\{\begin{array}{l}{\frac{\lambda_{1}}{2}\left\|\mathbf{w} \odot \mathbf{h}_{k}\right\|_{2}^{2}+} \\ {\frac{\mu}{2}\left\|\widehat{\mathbf{g}}_{k}-\sqrt{T} \mathbf{F} \mathbf{P}^{\top} \mathbf{h}_{k}+\widehat{\mathbf{s}}_{k}\right\|_{2}^{2}}\end{array}\right\}\\ \begin{array}{l}{=\left[\lambda_{1} \mathbf{W}^{\top} \mathbf{W}+\mu T \mathbf{P} \mathbf{P}^{T}\right]^{-1} \mu \sqrt T \mathbf{P}\mathbf F^{T}\left(\hat\mathbf{s}_{k}+\hat\mathbf{g}_{k}\right)} \\ {=\frac{\mu T \mathbf{P} \odot\left({\mathbf{s}_{k}}+\mathbf{g}_{k}\right)}{\lambda_{1}(\mathbf{w} \odot \mathbf{w})+\mu T }}\end{array}\\ \mathbf{s}_{k}=\frac{1}{\sqrt T}\mathbf F^{T}\hat\mathbf{s}_{k}\\\mathbf{g}_{k}=\frac{1}{\sqrt T}\mathbf F^{T}\hat\mathbf{g}_{k}$