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行向量对元素求导¶

设 $y^T=[y_1 ... y_n]$ 是n维行向量， $x$ 是元素

$\frac{\partial y^T}{\partial x}=[ \frac{\partial y_1}{\partial x} ... \frac{\partial y_n}{\partial x} ]$

列向量对元素求导¶

设 $y=\begin{bmatrix}y_1\\ ...\\ y_n\end{bmatrix}$ 是m维列向量， $x$ 是元素

则 $\frac{\partial y}{\partial x}=\begin{bmatrix} \frac{\partial y_1}{\partial x}\\ ... \\ \frac{\partial y_m}{\partial x} \end{bmatrix}$

矩阵对元素求导¶

设 $Y=\begin{bmatrix}y_1 & ... & y_{1n}\\...&...& ...\\ y_{m1} & ... & y_{mn}\end{bmatrix}$ 是 $m*n$ 维矩阵， $x$ 是元素

则 $\frac{\partial Y}{\partial x}=\begin{bmatrix} \frac{\partial y_{11}}{\partial x} & ... &\frac{\partial y_{1n}}{\partial x}\\...&...& ... \\ \frac{\partial y_{m1}}{\partial x}& ... & \frac{\partial y_{mn}}{\partial x} \end{bmatrix}$

元素对行向量求导¶

设 $y$ 是元素， $x^T=[x_1 ... x_q]$ 是q维行向量

则 $\frac{\partial y}{\partial x^T}=[ \frac{\partial y}{\partial x_1} ... \frac{\partial y}{\partial x_q} ]$

元素对列向量求导¶

设 $y$ 是元素， $x=\begin{bmatrix}x_1\\ ...\\ x_n\end{bmatrix}$ 是p维列向量

则 $\frac{\partial y}{\partial x}=\begin{bmatrix} \frac{\partial y}{\partial x_1}\\ ... \\ \frac{\partial y}{\partial x_p} \end{bmatrix}$

元素对矩阵求导¶

设 $y$ 是元素， $X=\begin{bmatrix}x_1 & ... & x_{1q}\\...&...& ...\\ x_{p1} & ... & x_{pq}\end{bmatrix}$ 是 $p*q$ 维矩阵

则 $\frac{\partial y}{\partial X}=\begin{bmatrix} \frac{\partial y}{\partial x_{11}} & ... &\frac{\partial y}{\partial x_{1q}}\\...&...& ... \\ \frac{\partial y}{\partial x_{p1}}& ... & \frac{\partial y}{\partial x_{pq}} \end{bmatrix}$

行向量对列向量求导¶

设 $y^T=[y_1 ... y_n]$ 是n维行向量， $x=\begin{bmatrix}x_1\\ ...\\ x_p\end{bmatrix}$ 是p维列向量

则 $\frac{\partial y^T}{\partial x}=\begin{bmatrix} \frac{\partial y_{1}}{\partial x_{1}} & ... &\frac{\partial y_{n}}{\partial x_{1}}\\...&...& ... \\ \frac{\partial y_1}{\partial x_{p}}& ... & \frac{\partial y_n}{\partial x_{p}} \end{bmatrix}$

列向量对行向量求导¶

设 $y=\begin{bmatrix}y_1\\ ...\\ y_m\end{bmatrix}$ 是m维列向量， $x^T=[x_1 ... x_q]$ 是q维行向量

则 $\frac{\partial y}{\partial x^T}=\begin{bmatrix} \frac{\partial y_{1}}{\partial x_{1}} & ... &\frac{\partial y_{1}}{\partial x_{q}}\\...&...& ... \\ \frac{\partial y_m}{\partial x_{1}}& ... & \frac{\partial y_m}{\partial x_{q}} \end{bmatrix}$

行向量对行向量求导¶

设 $y^T=[y_1 ... y_n]$ 是n维行向量， $x^T=[x_1 ... x_q]$ 是q维行向量

则 $\frac{\partial y^T}{\partial x^T}=[ \frac{\partial y^T}{\partial x_1} ... \frac{\partial y^T}{\partial x_q} ]$

列向量对列向量求导¶

设 $y=\begin{bmatrix}y_1\\ ...\\ y_m\end{bmatrix}$ 是m维列向量， $x=\begin{bmatrix}x_1\\ ...\\ x_p\end{bmatrix}$ 是p维列向量

则 $\frac{\partial y}{\partial x}=\begin{bmatrix} \frac{\partial y_1}{\partial x} \\...\\ \frac{\partial y_m}{\partial x} \end{bmatrix}$

矩阵对行向量求导¶

设 $Y=\begin{bmatrix}y_1 & ... & y_{1n}\\...&...& ...\\ y_{m1} & ... & y_{mn}\end{bmatrix}$ 是 $m*n$ 维矩阵, $x^T=[x_1 ... x_q]$ 是q维行向量

则 $\frac{\partial Y}{\partial x^T}=[ \frac{\partial Y}{\partial x_1} ... \frac{\partial Y}{\partial x_q} ]$

矩阵对列向量求导¶

设 $Y=\begin{bmatrix}y_1 & ... & y_{1n}\\...&...& ...\\ y_{m1} & ... & y_{mn}\end{bmatrix}$ 是 $m*n$ 维矩阵, $x=\begin{bmatrix}x_1\\ ...\\ x_n\end{bmatrix}$ 是p维列向量

行向量对矩阵求导¶

设 $y^T=[y_1 ... y_n]$ 是n维行向量， $X=\begin{bmatrix}x_1 & ... & x_{1q}\\...&...& ...\\ x_{p1} & ... & x_{pq}\end{bmatrix}$ 是 $p*q$ 维矩阵

则 $\frac{\partial y^T}{\partial X}=\begin{bmatrix} \frac{\partial y^T}{\partial x_{11}} & ... &\frac{\partial y^T}{\partial x_{1q}}\\...&...& ... \\ \frac{\partial y^T}{\partial x_{p1}}& ... & \frac{\partial y^T}{\partial x_{pq}} \end{bmatrix}$

列向量对矩阵求导¶

设 $y=\begin{bmatrix}y_1\\ ...\\ y_m\end{bmatrix}$ 是m维列向量， $X=\begin{bmatrix}x_1 & ... & x_{1q}\\...&...& ...\\ x_{p1} & ... & x_{pq}\end{bmatrix}$ 是 $p*q$ 维矩阵

则 $\frac{\partial y}{\partial X}=\begin{bmatrix} \frac{\partial y_1}{\partial X}\\ ... \\ \frac{\partial y_m}{\partial X} \end{bmatrix}$

矩阵对矩阵求导¶

设 $Y=\begin{bmatrix}y_1 & ... & y_{1n}\\...&...& ...\\ y_{m1} & ... & y_{mn}\end{bmatrix}=\begin{bmatrix}y_1^T\\ ...\\ y_m^T\end{bmatrix}$ 是 $m*n$ 维矩阵， $X=\begin{bmatrix}x_1 & ... & x_{1q}\\...&...& ...\\ x_{p1} & ... & x_{pq}\end{bmatrix}=[x_1 ... x_q]$ 是 $p*q$ 维矩阵

则 $\frac{\partial Y}{\partial X} =[ \frac{\partial Y}{\partial x_1} ... \frac{\partial Y}{\partial x_q} ] =\begin{bmatrix} \frac{\partial y_1^T}{\partial X}\\ ... \\ \frac{\partial y_m^T}{\partial X} \end{bmatrix} =\begin{bmatrix} \frac{\partial y_{1}^T}{\partial x_{1}} & ... &\frac{\partial y_{1}^T}{\partial x_{q}}\\ ...&...& ... \\ \frac{\partial y_m^T}{\partial x_{1}}& ... & \frac{\partial y_m^T}{\partial x_{q}} \end{bmatrix}$