常用矩阵求导公式速查

参考资料

• Matrix Calculu
• The Matrix Cookbook

常用矩阵求导公式

对于一个矩阵A，向量$\mathrm{x}$，有如下求导公式：
$$\frac{{\mathrm{dx}}^{\mathrm{T}}}{\mathrm{dx}}=I\text{,}\frac{\mathrm{dx}}{{\mathrm{dx}}^{\mathrm{T}}}=I\text{\hspace{1em}(1)}$$

d x T A d x = A , d A x d x T = A (2) \tag{2}

dxdxTA​=A,dxTdAx​=A​(2)

d A x d x = A T , d x A d x = A T (3) \tag{3}

dxdAx​=AT,dxdxA​=AT​(3)

d x T x d x = 2 x , d x T A x d x = ( A + A T ) x (4-1) \tag{4-1}

dxdxTx​=2x,dxdxTAx​=(A+AT)x​(4-1)
$$\frac{{\mathrm{dx}}^{\mathrm{T}}\mathrm{Ax}}{{\mathrm{dxx}}^{\mathrm{T}}}=\frac{d}{\mathrm{dx}}\left(\frac{{\mathrm{dx}}^{\mathrm{T}}\mathrm{Ax}}{\mathrm{dx}}\right)={\mathrm{A}}^{\mathrm{T}}+\mathrm{A}\text{\hspace{1em}(4-2)}$$

$$\frac{\partial \mathrm{u}}{\partial {\mathrm{x}}^{\mathrm{T}}}={\left(\frac{\partial {\mathrm{u}}^{\mathrm{T}}}{\partial \mathrm{x}}\right)}^{\mathrm{T}}\text{\hspace{1em}(5-1)}$$
∂ u T v ∂ x = ∂ u T ∂ x v + ∂ v T ∂ x u T , ∂ u v T ∂ x = ∂ u ∂ x v T + u ∂ v T ∂ x (5-2) \tag{5-2}

∂x∂uTv​=∂x∂uT​v+∂x∂vT​uT,∂x∂uvT​=∂x∂u​vT+u∂x∂vT​​(5-2)

$$\frac{\partial \left[(\mathrm{xu}-\mathrm{v}{)}^{\mathrm{T}}(\mathrm{x}u-\mathrm{v})\right]}{\partial \mathrm{x}}=2(\mathrm{xu}-\mathrm{v}){\mathrm{u}}^{\mathrm{T}}\text{\hspace{1em}(6)}$$

∂ u T x v ∂ x = u v T , ∂ u T x T x u ∂ x = 2 x u u T (7) \tag{7}

∂x∂uTxv​=uvT,∂x∂uTxTxu​=2xuuT​(7)

特别地，当$A=A^T$时，公式(4-1)和公式(4-2)有
d x T A x d x = 2 A x (8) \tag{8}

dxdxTAx​=2Ax​(8)
$$\frac{{\mathrm{dx}}^{\mathrm{T}}\mathrm{Ax}}{{\mathrm{dxx}}^{\mathrm{T}}}=2\mathrm{A}\text{\hspace{1em}(9)}$$