常用矩阵求导公式

常用矩阵求导公式

常见矩阵求导公式：

公式1

$$\frac{\text{d}{x}^T}{\text{d}x}=I\frac{\text{d}x}{\text{d}{x}^T}=I$$

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公式2

$$\frac{\text{d}{x}^{T}A}{\text{d}x}=A\frac{\text{d}Ax}{\text{d}{x}^T}=A$$

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公式3

$$\frac{\text{d}Ax}{\text{d}x}=A^T\frac{\text{d}xA}{\text{d}x}=A^T$$

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公式4

$$\frac{\partial u}{\partial x^T}={\left(\frac{\partial u^T}{\partial x}\right)}^T$$

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公式5

$$\frac{\partial u^{T}v}{\partial x}=\frac{\partial u^T}{\partial x}v+\frac{\partial v^T}{\partial x}{u}^T$$

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公式6
$$\frac{\partial u{v}^T}{\partial x}=\frac{\partial u}{\partial x}{v}^T+u\frac{\partial v^T}{\partial x}$$

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公式7

$$\frac{\text{d}{x}^{T}x}{\text{d}x}=2x$$
$$\frac{\text{d}{x}^{T}Ax}{\text{d}x}=\left(A+A^T\right)x$$

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公式8

$$\frac{\partial AB}{\partial x}=\frac{\partial A}{\partial x}B+A\frac{\partial B}{\partial x}$$

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公式9

$$\frac{\partial u^{T}Xv}{\partial X}=u{v}^T$$

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公式10

$$\frac{\partial u^T{X}^{T}Xu}{\partial X}=2Xu{u}^T$$

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公式11
$$\frac{\partial \left[{\left(Xu-v\right)}^T\left(Xu-v\right)\right]}{\partial X}=2\left(Xu-v\right)u^T$$

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