<>常用矩阵求导公式

常见矩阵求导公式:

公式1

d x T d x = I             d x d x T = I \frac{\text{d}x^T}{\text{d}x}=I\ \ \
\ \ \ \ \ \ \ \ \frac{\text{d}x}{\text{d}x^T}=IdxdxT​=I           dxTdx​=I

公式2

d x T A d x = A           d A x d x T = A \frac{\text{d}x^TA}{\text{d}x}=A\ \
\ \ \ \ \ \ \ \frac{\text{d}Ax}{\text{d}x^T}=AdxdxTA​=A         dxTdAx​=A

公式3

d A x d x = A T           d x A d x = A T \frac{\text{d}Ax}{\text{d}x}=A^T\ \
\ \ \ \ \ \ \ \frac{\text{d}xA}{\text{d}x}=A^TdxdAx​=AT         dxdxA​=AT

公式4

∂ u ∂ x T = ( ∂ u T ∂ x ) T \frac{\partial u}{\partial x^T}=\left(
\frac{\partial u^T}{\partial x} \right) ^T∂xT∂u​=(∂x∂uT​)T

公式5

∂ u T v ∂ x = ∂ u T ∂ x v + ∂ v T ∂ x u T \frac{\partial u^Tv}{\partial
x}=\frac{\partial u^T}{\partial x}v+\frac{\partial v^T}{\partial x}u^T∂x∂uTv​=∂x
∂uT​v+∂x∂vT​uT

公式6
∂ u v T ∂ x = ∂ u ∂ x v T + u ∂ v T ∂ x \frac{\partial uv^T}{\partial
x}=\frac{\partial u}{\partial x}v^T+u\frac{\partial v^T}{\partial x}∂x∂uvT​=∂x∂u
​vT+u∂x∂vT​

公式7

d x T x d x = 2 x \frac{\text{d}x^Tx}{\text{d}x}=2x dxdxTx​=2x
d x T A x d x = ( A + A T ) x \frac{\text{d}x^TAx}{\text{d}x}=\left( A+A^T
\right) xdxdxTAx​=(A+AT)x

公式8

∂ A B ∂ x = ∂ A ∂ x B + A ∂ B ∂ x \frac{\partial AB}{\partial
x}=\frac{\partial A}{\partial x}B+A\frac{\partial B}{\partial x}∂x∂AB​=∂x∂A​B+A∂
x∂B​

公式9

∂ u T X v ∂ X = u v T \frac{\partial u^TXv}{\partial X}=uv^T ∂X∂uTXv​=uvT

公式10

∂ u T X T X u ∂ X = 2 X u u T \frac{\partial u^TX^TXu}{\partial X}=2Xuu^T ∂X∂u
TXTXu​=2XuuT

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

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