深度学习-矩阵变换

矩阵求逆性质

$$(A^T{)}^T=A$$

$$(\lambda A{)}^T=\lambda A^T$$

$$(A\pm B{)}^T=A^T\pm B^T$$

$$(A\times B{)}^T=B^T\times A^T$$

矩阵求导性质

$$\frac{\partial AX}{\partial X}=A^T\Rightarrow \frac{\partial A^{T}X}{\partial X}=A$$

$$\frac{\partial XA}{\partial X}=A$$

$$\frac{\partial AXB}{\partial X}=A^T{B}^T\Rightarrow \frac{\partial AXB}{\partial X^T}=BA$$

$$\begin{gathered} \frac{\partial X^{T}AX}{\partial X}=(A+A^T)X \\ \Rightarrow \frac{\partial X^{T}A{A}^{T}X}{\partial X}=2A{A}^{T}X \\ \Rightarrow \frac{\partial X^{T}X}{\partial X}=\frac{\partial X^{T}E{E}^{T}X}{\partial X}=2E{E}^{T}X=2X \end{gathered}$$

维基百科：https://en.wikipedia.org/wiki/Matrix_calculus#Scalar-by-vector_identities

求导定义：求导法则.pdf wqmv

矩阵求导术上： https://zhuanlan.zhihu.com/p/24709748

矩阵求导术下：https://zhuanlan.zhihu.com/p/24863977