http://www.matrixcalculus.org/
标量（分子）分别对行/列向量（分母）各元素求导，结果仍为行/列向量（维度与分母一致）。
定义行向量：
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{{y}^{T}}=({{y}_{1}},{{y}_{2}},{{y}_{3}})
yT=(y1,y2,y3) ；列向量：
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x=(x1,x2,y3)T ；标量：
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标量对行向量求导：
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\frac{dc}{d{{y}^{T}}}=\left( \begin{matrix} \frac{dc}{d{{y}_{1}}} & \frac{dc}{d{{y}_{2}}} & \frac{dc}{d{{y}_{3}}} \\ \end{matrix} \right)
dyTdc=(dy1dcdy2dcdy3dc)
标量对列向量求导：
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\frac{dc}{dx}={{\left( \begin{matrix} \frac{dc}{d{{x}_{1}}} & \frac{dc}{d{{x}_{2}}} & \frac{dc}{d{{x}_{3}}} \\ \end{matrix} \right)}^{T}}
dxdc=(dx1dcdx2dcdx3dc)T
向量（分子）的各个元素分别对行/列向量（分母）各元素求导，结果仍为行/列向量（维度与分母一致），再将各个行/列向量按照分子向量的维度排列。
定义1×m维行向量：
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{{y}^{T}}=\left( \begin{matrix} {{y}_{1}} & {{y}_{2}} & \cdots & {{y}_{m}} \\ \end{matrix} \right)
yT=(y1y2⋯ym) ；n×1维列向量：
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x={{\left( \begin{matrix} {{x}_{1}} & {{x}_{2}} & \cdots & {{x}_{n}} \\ \end{matrix} \right)}^{T}}
x=(x1x2⋯xn)T ；
行向量对列向量求导：
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\frac{d{{y}^{T}}}{dx}=\left( \begin{matrix} \frac{d{{y}_{1}}}{d{{x}_{1}}} & \frac{d{{y}_{2}}}{d{{x}_{1}}} & \cdots & \frac{d{{y}_{m}}}{d{{x}_{1}}} \\ \frac{d{{y}_{1}}}{d{{x}_{2}}} & \frac{d{{y}_{2}}}{d{{x}_{2}}} & \cdots & \frac{d{{y}_{m}}}{d{{x}_{2}}} \\ \vdots & \vdots & \cdots & \vdots \\ \frac{d{{y}_{1}}}{d{{x}_{n}}} & \frac{d{{y}_{2}}}{d{{x}_{n}}} & \cdots & \frac{d{{y}_{m}}}{d{{x}_{n}}} \\ \end{matrix} \right)
dxdyT=⎝⎜⎜⎜⎜⎛dx1dy1dx2dy1⋮dxndy1dx1dy2dx2dy2⋮dxndy2⋯⋯⋯⋯dx1dymdx2dym⋮dxndym⎠⎟⎟⎟⎟⎞
列向量对行向量求导：
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\frac{dy}{d{{x}^{T}}}=\left( \begin{matrix} \frac{d{{y}_{1}}}{d{{x}_{1}}} & \frac{d{{y}_{1}}}{d{{x}_{2}}} & \cdots & \frac{d{{y}_{1}}}{d{{x}_{n}}} \\ \frac{d{{y}_{2}}}{d{{x}_{1}}} & \frac{d{{y}_{2}}}{d{{x}_{2}}} & \cdots & \frac{d{{y}_{2}}}{d{{x}_{n}}} \\ \vdots & \vdots & \cdots & \vdots \\ \frac{d{{y}_{m}}}{d{{x}_{1}}} & \frac{d{{y}_{m}}}{d{{x}_{2}}} & \cdots & \frac{d{{y}_{m}}}{d{{x}_{n}}} \\ \end{matrix} \right)
dxTdy=⎝⎜⎜⎜⎜⎛dx1dy1dx1dy2⋮dx1dymdx2dy1dx2dy2⋮dx2dym⋯⋯⋯⋯dxndy1dxndy2⋮dxndym⎠⎟⎟⎟⎟⎞
由上述两个公式可知：
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\frac{d{{y}^{T}}}{dx}={{\left( \frac{dy}{d{{x}^{T}}} \right)}^{T}}
dxdyT=(dxTdy)T
行向量对行向量求导：
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\frac{d{{y}^{T}}}{d{{x}^{T}}}=\left( \begin{matrix} \frac{d{{y}_{1}}}{d{{x}_{1}}} & \cdots & \frac{d{{y}_{1}}}{d{{x}_{n}}} & \frac{d{{y}_{2}}}{d{{x}_{1}}} & \cdots & \frac{d{{y}_{2}}}{d{{x}_{m}}} & \cdots & \frac{d{{y}_{m}}}{d{{x}_{1}}} & \cdots & \frac{d{{y}_{m}}}{d{{x}_{n}}} \\ \end{matrix} \right)
dxTdyT=(dx1dy1⋯dxndy1dx1dy2⋯dxmdy2⋯dx1dym⋯dxndym)
列向量对列向量求导：
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\frac{dy}{dx}={{\left( \begin{matrix} \frac{d{{y}_{1}}}{d{{x}_{1}}} & \cdots & \frac{d{{y}_{1}}}{d{{x}_{n}}} & \frac{d{{y}_{2}}}{d{{x}_{1}}} & \cdots & \frac{d{{y}_{2}}}{d{{x}_{m}}} & \cdots & \frac{d{{y}_{m}}}{d{{x}_{1}}} & \cdots & \frac{d{{y}_{m}}}{d{{x}_{n}}} \\ \end{matrix} \right)}^{T}}
dxdy=(dx1dy1⋯dxndy1dx1dy2⋯dxmdy2⋯dx1dym⋯dxndym)T
雅可比矩阵定义：假设
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F:{{\mathbf{R}}_{n}}\to {{\mathbf{R}}_{m}}
F:Rn→Rm是一个将n维欧氏空间映射到m维欧氏空间的函数。这个函数由m个实函数组成：
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{{y}_{1}}({{x}_{1}},{{x}_{2}},\cdots ,{{x}_{n}})
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{{y}_{2}}({{x}_{1}},{{x}_{2}},\cdots ,{{x}_{n}})
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{{y}_{m}}({{x}_{1}},{{x}_{2}},\cdots ,{{x}_{n}})
ym(x1,x2,⋯,xn) 。这些函数的偏导数(如果存在)可以组成一个m行n列的矩阵，这个矩阵就是所谓的雅可比矩阵：
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\left( \begin{matrix} \frac{d{{y}_{1}}}{d{{x}_{1}}} & \frac{d{{y}_{1}}}{d{{x}_{2}}} & \cdots & \frac{d{{y}_{1}}}{d{{x}_{n}}} \\ \frac{d{{y}_{2}}}{d{{x}_{1}}} & \frac{d{{y}_{2}}}{d{{x}_{2}}} & \cdots & \frac{d{{y}_{2}}}{d{{x}_{n}}} \\ \vdots & \vdots & \cdots & \vdots \\ \frac{d{{y}_{m}}}{d{{x}_{1}}} & \frac{d{{y}_{m}}}{d{{x}_{2}}} & \cdots & \frac{d{{y}_{m}}}{d{{x}_{n}}} \\ \end{matrix} \right)
⎝⎜⎜⎜⎜⎛dx1dy1dx1dy2⋮dx1dymdx2dy1dx2dy2⋮dx2dym⋯⋯⋯⋯dxndy1dxndy2⋮dxndym⎠⎟⎟⎟⎟⎞，记作
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{{J}_{F}}({{x}_{1}},{{x}_{2}},\cdots ,{{x}_{n}})
JF(x1,x2,⋯,xn)或
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\frac{\partial ({{y}_{1}},{{y}_{2}},\cdots ,{{y}_{m}})}{\partial ({{x}_{1}},{{x}_{2}},\cdots ,{{x}_{n}})}
∂(x1,x2,⋯,xn)∂(y1,y2,⋯,ym)。这个矩阵的第i行是
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{{y}_{i}}(i=1,2,\cdots m)
yi(i=1,2,⋯m)的梯度函数的转置。
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{{\mathbf{R}}_{3}}\to {{\mathbf{R}}_{4}}
R3→R4的一个函数：
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{y_1} = {x_1},{y_2} = 5{x_3},{y_3} = 4x_2^2 - 2{x_3},{y_4} = {x_3}\sin {x_1}
y1=x1,y2=5x3,y3=4x22−2x3,y4=x3sinx1，它的雅可比矩阵为：
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{{J}_{F}}=\left( \begin{matrix} \frac{d{{y}_{1}}}{d{{x}_{1}}} & \frac{d{{y}_{1}}}{d{{x}_{2}}} & \frac{d{{y}_{1}}}{d{{x}_{3}}} \\ \frac{d{{y}_{2}}}{d{{x}_{1}}} & \frac{d{{y}_{2}}}{d{{x}_{2}}} & \frac{d{{y}_{2}}}{d{{x}_{3}}} \\ \frac{d{{y}_{3}}}{d{{x}_{1}}} & \frac{d{{y}_{3}}}{d{{x}_{2}}} & \frac{d{{y}_{3}}}{d{{x}_{3}}} \\ \frac{d{{y}_{4}}}{d{{x}_{1}}} & \frac{d{{y}_{4}}}{d{{x}_{2}}} & \frac{d{{y}_{4}}}{d{{x}_{3}}} \\ \end{matrix} \right)=\left( \begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 5 \\ 0 & 8{{x}_{2}} & -2 \\ {{x}_{3}}\cos {{x}_{1}} & 0 & \sin {{x}_{1}} \\ \end{matrix} \right)
JF=⎝⎜⎜⎜⎛dx1dy1dx1dy2dx1dy3dx1dy4dx2dy1dx2dy2dx2dy3dx2dy4dx3dy1dx3dy2dx3dy3dx3dy4⎠⎟⎟⎟⎞=⎝⎜⎜⎛100x3cosx1008x2005−2sinx1⎠⎟⎟⎞
有关雅可比矩阵的详细内容请参考网址：http://jacoxu.com/jacobian%E7%9F%A9%E9%98%B5%E5%92%8Chessian%E7%9F%A9%E9%98%B5/.
定义m×1维列向量：
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u={{\left( \begin{matrix} {{u}_{1}} & {{u}_{2}} & \cdots & {{u}_{m}} \\ \end{matrix} \right)}^{T}}
u=(u1u2⋯um)T ，
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v={{\left( \begin{matrix} {{v}_{1}} & {{v}_{2}} & \cdots & {{v}_{m}} \\ \end{matrix} \right)}^{T}}
v=(v1v2⋯vm)T ；
定义n×1维列向量：
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w={{\left( \begin{matrix} {{w}_{1}} & {{w}_{2}} & \cdots & {{w}_{n}} \\ \end{matrix} \right)}^{T}}
w=(w1w2⋯wn)T
点乘对列向量求导：
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\frac{{d\left( {u \cdot v} \right)}}{{dw}} = \frac{{d\left( {{v^T}u} \right)}}{{dw}} = \frac{{d{u^T}}}{{dw}} \cdot v + \frac{{d{v^T}}}{{dw}} \cdot u
dwd(u⋅v)=dwd(vTu)=dwduT⋅v+dwdvT⋅u
例1：
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\frac{d\left( u\cdot v \right)}{du}=\frac{d\left( {{v}^{T}}u \right)}{du}=\frac{d\sum{{{u}_{i}}{{v}_{i}}}}{du}={{\left( \begin{matrix} \frac{d\sum{{{u}_{i}}{{v}_{i}}}}{d{{u}_{1}}} & \frac{d\sum{{{u}_{i}}{{v}_{i}}}}{d{{u}_{2}}} & \cdots & \frac{d\sum{{{u}_{i}}{{v}_{i}}}}{d{{u}_{m}}} \\ \end{matrix} \right)}^{T}}={{\left( \begin{matrix} {{v}_{1}} & {{v}_{2}} & \cdots & {{v}_{k}} \\ \end{matrix} \right)}^{T}}=v
dud(u⋅v)=dud(vTu)=dud∑uivi=(du1d∑uividu2d∑uivi⋯dumd∑uivi)T=(v1v2⋯vk)T=v
例2：
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\frac{d\left( u\cdot v \right)}{dv}=\frac{d\left( {{v}^{T}}u \right)}{dv}=\frac{d\sum{{{u}_{i}}{{v}_{i}}}}{dv}={{\left( \begin{matrix} \frac{d\sum{{{u}_{i}}{{v}_{i}}}}{d{{v}_{1}}} & \frac{d\sum{{{u}_{i}}{{v}_{i}}}}{d{{v}_{2}}} & \cdots & \frac{d\sum{{{u}_{i}}{{v}_{i}}}}{d{{v}_{m}}} \\ \end{matrix} \right)}^{T}}={{\left( \begin{matrix} {{u}_{1}} & {{u}_{2}} & \cdots & {{u}_{k}} \\ \end{matrix} \right)}^{T}}=u
dvd(u⋅v)=dvd(vTu)=dvd∑uivi=(dv1d∑uividv2d∑uivi⋯dvmd∑uivi)T=(u1u2⋯uk)T=u
例3：
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\frac{d\left( {{x}^{T}}x \right)}{dy}=2\frac{d\left( {{x}^{T}} \right)}{dy}x
dyd(xTx)=2dyd(xT)x
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\frac{d\left( {{x}^{T}}\cdot x \right)}{dx}=2x
dxd(xT⋅x)=2x
叉乘对列向量求导：（以3×1的列向量为例，即m=n=3）
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\frac{d{{\left( u\times v \right)}^{T}}}{dw}=\frac{d\left( \begin{matrix} {{u}_{2}}{{v}_{3}}-{{u}_{3}}{{v}_{2}} & {{u}_{3}}{{v}_{1}}-{{u}_{1}}{{v}_{3}} & {{u}_{1}}{{v}_{2}}-{{u}_{2}}{{v}_{1}} \\ \end{matrix} \right)}{dw}
dwd(u×v)T=dwd(u2v3−u3v2u3v1−u1v3u1v2−u2v1)
例1：
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\frac{d{{\left( u\times v \right)}^{T}}}{du}=\left( \begin{matrix} 0 & -{{v}_{3}} & {{v}_{2}} \\ {{v}_{3}} & 0 & -{{v}_{1}} \\ -{{v}_{2}} & {{v}_{1}} & 0 \\ \end{matrix} \right)
dud(u×v)T=⎝⎛0v3−v2−v30v1v2−v10⎠⎞
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\frac{d{{\left( u\times v \right)}^{T}}}{du}w=\left( \begin{matrix} 0 & -{{v}_{3}} & {{v}_{2}} \\ {{v}_{3}} & 0 & -{{v}_{1}} \\ -{{v}_{2}} & {{v}_{1}} & 0 \\ \end{matrix} \right)\left( \begin{matrix} {{w}_{1}} \\ {{w}_{2}} \\ {{w}_{3}} \\ \end{matrix} \right)=\left( \begin{matrix} -{{v}_{3}}{{w}_{2}}+{{v}_{2}}{{w}_{3}} \\ {{v}_{3}}{{w}_{1}}-{{v}_{1}}{{w}_{3}} \\ -{{v}_{2}}{{w}_{1}}+{{v}_{1}}{{w}_{2}} \\ \end{matrix} \right)=v\times w
dud(u×v)Tw=⎝⎛0v3−v2−v30v1v2−v10⎠⎞⎝⎛w1w2w3⎠⎞=⎝⎛−v3w2+v2w3v3w1−v1w3−v2w1+v1w2⎠⎞=v×w
例2：
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\frac{d{{\left( u\times v \right)}^{T}}}{dv}=\left( \begin{matrix} 0 & {{u}_{3}} & -{{u}_{2}} \\ -{{u}_{3}} & 0 & {{u}_{1}} \\ {{u}_{2}} & -{{u}_{1}} & 0 \\ \end{matrix} \right)
dvd(u×v)T=⎝⎛0−u3u2u30−u1−u2u10⎠⎞
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\frac{d{{\left( u\times v \right)}^{T}}}{dv}w=\left( \begin{matrix} 0 & {{u}_{3}} & -{{u}_{2}} \\ -{{u}_{3}} & 0 & {{u}_{1}} \\ {{u}_{2}} & -{{u}_{1}} & 0 \\ \end{matrix} \right)\left( \begin{matrix} {{w}_{1}} \\ {{w}_{2}} \\ {{w}_{3}} \\ \end{matrix} \right)=\left( \begin{matrix} {{u}_{3}}{{w}_{2}}-{{u}_{2}}{{w}_{3}} \\ -{{u}_{3}}{{w}_{1}}+{{u}_{1}}{{w}_{3}} \\ {{u}_{2}}{{w}_{1}}-{{u}_{1}}{{w}_{2}} \\ \end{matrix} \right)=w\times u
dvd(u×v)Tw=⎝⎛0−u3u2u30−u1−u2u10⎠⎞⎝⎛w1w2w3⎠⎞=⎝⎛u3w2−u2w3−u3w1+u1w3u2w1−u1w2⎠⎞=w×u
例3：向量的叉乘对向量求导，即雅可比矩阵
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\boldsymbol{J}=\frac{\partial(\boldsymbol{u} \times \boldsymbol{v})}{\partial \boldsymbol{u}}=\frac{\partial}{\partial \boldsymbol{u}}\left[\begin{array}{c}-u_{3} v_{2}+u_{2} v_{3} \\ u_{3} v_{1}-u_{1} v_{3} \\ -u_{2} v_{1}+u_{1} v_{2}\end{array}\right]=\left[\begin{array}{ccc}0 & v_{3} & -v_{2} \\ -v_{3} & 0 & v_{1} \\ v_{2} & -v_{1} & 0\end{array}\right]=-\boldsymbol{v}^{\times}
J=∂u∂(u×v)=∂u∂⎣⎡−u3v2+u2v3u3v1−u1v3−u2v1+u1v2⎦⎤=⎣⎡0−v3v2v30−v1−v2v10⎦⎤=−v×
例4：向量的叉乘对向量求导，即雅可比矩阵
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\boldsymbol{J}=\frac{\partial(\boldsymbol{u} \times \boldsymbol{v})}{\partial \boldsymbol{v}}=\frac{\partial}{\partial \boldsymbol{v}}\left[\begin{array}{c}-u_{3} v_{2}+u_{2} v_{3} \\ u_{3} v_{1}-u_{1} v_{3} \\ -u_{2} v_{1}+u_{1} v_{2}\end{array}\right]=\left[\begin{array}{ccc}0 & -u_{3} & u_{2} \\ u_{3} & 0 & -u_{1} \\ -u_{2} & u_{1} & 0\end{array}\right]=\boldsymbol{u}^{\times}
J=∂v∂(u×v)=∂v∂⎣⎡−u3v2+u2v3u3v1−u1v3−u2v1+u1v2⎦⎤=⎣⎡0u3−u2−u30u1u2−u10⎦⎤=u×
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\frac{d{{\left( {u}/{\left\| u \right\|}^n\; \right)}^{T}}}{du}=\left( \begin{matrix} \frac{d\left( \frac{{{u}_{1}}}{\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}^n} \right)}{d{{u}_{1}}} & \frac{d\left( \frac{{{u}_{2}}}{\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}^n} \right)}{d{{u}_{1}}} & \frac{d\left( \frac{{{u}_{3}}}{\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}^n} \right)}{d{{u}_{1}}} \\ \frac{d\left( \frac{{{u}_{1}}}{\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}^n} \right)}{d{{u}_{2}}} & \frac{d\left( \frac{{{u}_{2}}}{\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}^n} \right)}{d{{u}_{2}}} & \frac{d\left( \frac{{{u}_{3}}}{\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}^n} \right)}{d{{u}_{2}}} \\ \frac{d\left( \frac{{{u}_{1}}}{\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}^n} \right)}{d{{u}_{3}}} & \frac{d\left( \frac{{{u}_{2}}}{\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}^n} \right)}{d{{u}_{3}}} & \frac{d\left( \frac{{{u}_{3}}}{\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}^n} \right)}{d{{u}_{3}}} \\ \end{matrix} \right)=\left( \begin{matrix} \frac{1}{{\left\| u \right\|}^n}-\frac{u_{1}^{2}}{{{\left\| u \right\|}^{n+2}}} & -\frac{{{u}_{2}}{{u}_{1}}}{{{\left\| u \right\|}^{n+2}}} & -\frac{{{u}_{3}}{{u}_{1}}}{{{\left\| u \right\|}^{n+2}}} \\ -\frac{{{u}_{1}}{{u}_{2}}}{{{\left\| u \right\|}^{n+2}}} & \frac{1}{{\left\| u \right\|}^n}-\frac{u_{2}^{2}}{{{\left\| u \right\|}^{3}}} & -\frac{{{u}_{3}}{{u}_{2}}}{{{\left\| u \right\|}^{n+2}}} \\ -\frac{{{u}_{1}}{{u}_{3}}}{{{\left\| u \right\|}^{n+2}}} & -\frac{{{u}_{2}}{{u}_{3}}}{{{\left\| u \right\|}^{n+2}}} & \frac{1}{{\left\| u \right\|}^n}-\frac{u_{3}^{2}}{{{\left\| u \right\|}^{n+2}}} \\ \end{matrix} \right)=\frac{{{I}_{3}}}{{\left\| u \right\|}^n}-\frac{u\cdot {{u}^{T}}}{{{\left\| u \right\|}^{n+2}}}
dud(u/∥u∥n)T=⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎛du1d(u12+u22+u32
nu1)du2d(u12+u22+u32
nu1)du3d(u12+u22+u32
nu1)du1d(u12+u22+u32
nu2)du2d(u12+u22+u32
nu2)du3d(u12+u22+u32
nu2)du1d(u12+u22+u32
nu3)du2d(u12+u22+u32
nu3)du3d(u12+u22+u32
nu3)⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎞=⎝⎜⎜⎛∥u∥n1−∥u∥n+2u12−∥u∥n+2u1u2−∥u∥n+2u1u3−∥u∥n+2u2u1∥u∥n1−∥u∥3u22−∥u∥n+2u2u3−∥u∥n+2u3u1−∥u∥n+2u3u2∥u∥n1−∥u∥n+2u32⎠⎟⎟⎞=∥u∥nI3−∥u∥n+2u⋅uT
