矩阵迹的求导

关于矩阵迹的一些求导法则

一、迹的基本性质

• 线性性质：$tr(\sum_ic_iA_i)=\sum_ic_itr(A_i)$
• 转置不变性：$tr(A)=tr(A^T)$
• 轮换不变性 $tr(ABC)=tr(BCA)=tr(CAB)$

注意：轮换不变性不等于交换性，例如$tr(ABC)\neq tr(ACB)$

二、范数与迹的关系

1. $\Vert X\Vert_\mathcal{F}^2=tr(X^TX)=\sum_{i=1}^m\sum_{j=1}^na_{ij}^2$

三、迹的求导规则

1. $\dfrac{\partial tr(AB)}{\partial A}=\dfrac{\partial tr(BA)}{\partial A}=B^T$

2. $\dfrac{\partial tr(A^TB)}{\partial A}=\dfrac{\partial tr(BA^T)}{\partial A}=B$

3. $\dfrac{\partial tr(A^TBA)}{\partial A}=BA+B^TA$

4. $\dfrac{\partial (ABA^T)}{\partial A}=AB^T+AB$

5. $\dfrac{\partial tr(AXBXC^T)}{\partial X}=A^TCX^TB^T+B^TX^TA^TC$

6. $\dfrac{\partial tr(AXBX)}{\partial X}=A^TX^TB^T+B^TX^TA^T$

7. $\dfrac{\partial tr(AXBX^T)}{\partial X}=A^TXB^T+AXB​$

8. $\dfrac{\partial tr(A^TXB^T)}{\partial X}=\dfrac{\partial tr(AX^TB)}{\partial X}=AB$

9. $\dfrac{\partial tr(A^TXB)}{\partial X}=AB^T$

10. $\dfrac{\partial tr(A^TX^TXA)}{\partial X}=2XAA^T$

11. $\dfrac{\partial tr[(XA-B)^T(XA-B)]}{\partial X}=2(XA-B)A^T$

    核心公式：$\nabla tr(XAX^TB)=B^TXA^T+BXA$