线性代数 - 1 - 基础知识

约定不等于承诺〃 2022-11-13 14:25 469阅读 0赞

> 线性代数，基础知识，温故知新。

### 定义 ###

*  向量：

> 向量默认为列向量：

x → = ( x 1 , x 2 , ⋯   , x n ) T = \[ x 1 x 2 ⋮ x n \] \\overrightarrow\{\\mathbf\{x\}\}=\\left(x\_\{1\}, x\_\{2\}, \\cdots, x\_\{n\}\\right)^\{T\}=\\left\[\\begin\{array\}\{c\} x\_\{1\} \\\\ x\_\{2\} \\\\ \\vdots \\\\ x\_\{n\} \\end\{array\}\\right\] x=(x1,x2,⋯,xn)T=⎣⎢⎢⎢⎡x1x2⋮xn⎦⎥⎥⎥⎤

*  矩阵  X ∈ R m × n \\mathbf\{X\} \\in \\mathbb\{R\}^\{m \\times n\} X∈Rm×n，表示为：

X = \[ x 1 , 1 x 1 , 2 ⋯ x 1 , n x 2 , 1 x 2 , 2 ⋯ x 2 , n ⋮ ⋮ ⋱ ⋮ x m , 1 x m , 2 ⋯ x m , n \] \\mathbf\{X\}=\\left\[\\begin\{array\}\{cccc\} x\_\{1,1\} & x\_\{1,2\} & \\cdots & x\_\{1, n\} \\\\ x\_\{2,1\} & x\_\{2,2\} & \\cdots & x\_\{2, n\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ x\_\{m, 1\} & x\_\{m, 2\} & \\cdots & x\_\{m, n\} \\end\{array\}\\right\] X=⎣⎢⎢⎢⎡x1,1x2,1⋮xm,1x1,2x2,2⋮xm,2⋯⋯⋱⋯x1,nx2,n⋮xm,n⎦⎥⎥⎥⎤

### 范数 ###

#### 向量范数 ####

##### **1-范数** #####

> 各个元素的绝对值之和

∥ X ∥ 1 = ∑ i = 1 n ∣ x i ∣ \\|X\\|\_\{1\}=\\sum\_\{i=1\}^\{n\}\\left|x\_\{i\}\\right| ∥X∥1=i=1∑n∣xi∣

##### 2-范数 #####

> 每个元素的平方和再开平方根

∥ X ∥ 2 = ( ∑ i = 1 n x i 2 ) 1 2 = ∑ i = 1 n x i 2 \\|X\\|\_\{2\}=\\left(\\sum\_\{i=1\}^\{n\} x\_\{i\}^\{2\}\\right)^\{\\frac\{1\}\{2\}\}=\\sqrt\{\\sum\_\{i=1\}^\{n\} x\_\{i\}^\{2\}\} ∥X∥2=(i=1∑nxi2)21=i=1∑nxi2

##### **p-范数** #####

∥ X ∥ p = ( ∑ i = 1 n ∣ x i ∣ p ) 1 p \\|X\\|\_\{p\}=\\left(\\sum\_\{i=1\}^\{n\}\\left|x\_\{i\}\\right|^\{p\}\\right)^\{\\frac\{1\}\{p\}\} ∥X∥p=(i=1∑n∣xi∣p)p1

*  其中正整数*p*≥1，并且有\{% raw %\}  lim ⁡ p → ∞ ∥ X ∥ p = max ⁡ 1 ≤ i ≤ n ∣ x i ∣ \\lim \_\{p \\rightarrow \\infty\}\\|X\\|\_\{p\}=\\max \_\{1 \\leq i \\leq n\}\\left|x\_\{i\}\\right| limp→∞∥X∥p=max1≤i≤n∣xi∣.\{% endraw %\}

##### **无穷范数** #####

∥ X ∥ ∞ = max ⁡ 1 < i < n ∣ x i ∣ \\|X\\|\_\{\\infty\}=\\max \_ \{ 1 < i< n\}\\left|x\_\{i\}\\right | ∥X∥∞=1<i<nmax∣xi∣

> 为向量中绝对值最大的元素的值。

#### 矩阵范数 ####

##### **1-范数（列模）** #####

> 矩阵的每一列上的元素绝对值先求和，再从中取个最大的，（列和最大）  
> \{% raw %\}

∥ A ∥ 1 = max ⁡ 1 ≤ j ≤ n ∑ i = 1 n ∣ a i j ∣ \\|A\\|\_\{1\}=\\max \_\{1 \\leq j \\leq n\} \\sum\_\{i=1\}^\{n\} \\mid a\_\{i j\}\\mid ∥A∥1=1≤j≤nmaxi=1∑n∣aij∣

##### **2-范数（谱模）：** #####

> 最大特征值开平方根：  
> \{% raw %\}

∣ ∣ A ∥ 2 = λ max ⁡ ( A T A ) = max ⁡ 1 ≤ i ≤ n ∣ λ i ∣ \\mid\\mid A \\|\_\{2\}=\\sqrt\{\\lambda\_\{\\max \}\\left(A^\{T\} A\\right)\}=\\sqrt\{\\max \_\{1 \\leq i \\leq n\}\\left|\\lambda\_\{i\}\\right|\} ∣∣A∥2=λmax(ATA)=1≤i≤nmax∣λi∣

##### **无穷范数（行模）**： #####

> 矩阵的每一行上的元素绝对值先求和，再从中取个最大的，（行和最大）

∥ A ∥ ∞ = max ⁡ 1 ≤ i ≤ n ∑ j = 1 n ∣ a i j ∣ \\|A\\|\_\{\\infty\}=\\max \_\{1 \\leq i \\leq n\} \\sum\_\{j=1\}^\{n\} \\mid a\_\{i j\}\\mid ∥A∥∞=1≤i≤nmaxj=1∑n∣aij∣

##### **L0范数**： #####

> 矩阵的非0元素的个数，通常用它来表示稀疏，L0范数越小0元素越多，也就越稀疏

##### **L1范数：** #####

> 矩阵中的每个元素绝对值之和，它是L0范数的最优凸近似，因此它也可以近似表示稀疏

##### **F范数：** #####

> 矩阵的各个元素平方之和再开平方根，它通常也叫做矩阵的L2范数，它的优点在它是一个凸函数，可以求导求解，易于计算

∥ A ∥ F = ∑ i , j a i , j 2 \\|\\mathbf\{A\}\\|\_\{F\}=\\sqrt\{\\sum\_\{i, j\} a\_\{i, j\}^\{2\}\} ∥A∥F=i,j∑ai,j2

### 行列式 ###

*  方阵  A A A 的行列式，记作  d e t ( A ) det(A) det(A)或 ∣ A ∣ |A| ∣A∣：

D = ∣ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋯ ⋯ ⋯ ⋯ a n 1 a n 2 ⋯ a n n ∣ D=\\left|\\begin\{array\}\{llll\} a\_\{11\} & a\_\{12\} & \\cdots & a\_\{1 n\} \\\\ a\_\{21\} & a\_\{22\} & \\cdots & a\_\{2 n\} \\\\ \\cdots & \\cdots & \\cdots & \\cdots \\\\ a\_\{n 1\} & a\_\{n 2\} & \\cdots & a\_\{n n\} \\end\{array\}\\right| D=∣∣∣∣∣∣∣∣a11a21⋯an1a12a22⋯an2⋯⋯⋯⋯a1na2n⋯ann∣∣∣∣∣∣∣∣

*  计算公式：

D = ∑ ( − 1 ) k a 1 k 1 a 2 k 2 ⋯ a n k n D=\\sum (- 1) ^ \{k \} a\_\{1 k\_\{ 1\} \} a\_\{2 k\_\{2\} \} \\cdots a\_ \{ n k \_ \{ n\} \} D=∑(−1)ka1k1a2k2⋯ankn

*  表示的是 n n n个 n n n维向量构成的n维平行多面体的体积，该体积有正负，若存在线性相关的向量，行列式为0
 *  行列式 A A A中某行(或列)用同一数 k k k乘,其结果等于 k A kA kA
 *  行列式 A A A等于其转置行列式 A T A^T AT( A T A^T AT的第i行为 A A A的第 i i i列)
 *  行列式 A A A中两行（或列）互换,其结果等于 − A -A −A
 *  把行列式 A A A的某行（或列）中各元同乘一数后加到另一行（或列）中各对应元上，结果仍然是 A A A

### 方阵的迹 ###

*  方阵 A = ( a i , j ) n × n \\mathbf\{A\}=\\left(a\_\{i, j\}\\right)\_\{n \\times n\} A=(ai,j)n×n的迹，记作 tr ⁡ ( A ) \\operatorname\{tr\}(\\mathbf\{A\}) tr(A)，对角线元素之和，也等于特征值的和：

tr ⁡ ( A ) = ∑ i a i , i \\operatorname\{tr\}(\\mathbf\{A\})=\\sum\_\{i\} a\_\{i, i\} tr(A)=i∑ai,i

### 向量积 ###

#### 点积（Dot Product） ####

> 对应元素乘积和，结果不是一个向量，而是一个标量（Scalar）

A ⋅ B = ∑ i a i b i = ∣ A ∣ ∣ B ∣ C o s ( θ ) \{\\bf\{A\} \} \\cdot \{\\bf\{B\} \} = \\sum\\limits\_i \{ \{a\_i\}\{b\_i\} \} = \\left| A \\right|\\left| B \\right|Cos\\left( \\theta \\right) A⋅B=i∑aibi=∣A∣∣B∣Cos(θ)

#### **叉乘（cross product）** ####

##### 三维向量的叉积： #####

> 用三阶行列式表示

u → = u x i → + u y j → + u z k → , v → = v x i → + v y j → + v z k → \\overrightarrow\{\\mathbf\{u\}\}=u\_\{x\} \\overrightarrow\{\\mathbf\{i\}\}+u\_\{y\} \\overrightarrow\{\\mathbf\{j\}\}+u\_\{z\} \\overrightarrow\{\\mathbf\{k\}\}, \\quad \\overrightarrow\{\\mathbf\{v\}\}=v\_\{x\} \\overrightarrow\{\\mathbf\{i\}\}+v\_\{y\} \\overrightarrow\{\\mathbf\{j\}\}+v\_\{z\} \\overrightarrow\{\\mathbf\{k\}\} u=uxi\+uyj\+uzk,v=vxi\+vyj\+vzk

> 其中 i → , j → , k → \\overrightarrow\{\\mathbf\{i\}\}, \\overrightarrow\{\\mathbf\{j\}\}, \\overrightarrow\{\\mathbf\{k\}\} i,j,k分别为 x , y , z x, y, z x,y,z轴的单位向量。

w → = u → × v → = \[ i → j → k → u x u y u z v x v y v z \] \\overrightarrow\{\\mathbf\{w\}\}=\\overrightarrow\{\\mathbf\{u\}\} \\times \\overrightarrow\{\\mathbf\{v\}\}=\\left\[\\begin\{array\}\{ccc\} \\overrightarrow\{\\mathbf\{i\}\} & \\overrightarrow\{\\mathbf\{j\}\} & \\overrightarrow\{\\mathbf\{k\}\} \\\\ u\_\{x\} & u\_\{y\} & u\_\{z\} \\\\ v\_\{x\} & v\_\{y\} & v\_\{z\} \\end\{array\}\\right\] w=u×v=⎣⎡iuxvxjuyvykuzvz⎦⎤

*   u → , v → \\overrightarrow\{\\mathbf\{u\}\}, \\overrightarrow\{\\mathbf\{v\}\} u,v的叉积垂直于 u → , v → \\overrightarrow\{\\mathbf\{u\}\}, \\overrightarrow\{\\mathbf\{v\}\} u,v构成的平面，其方向符合右手规则
 *  叉积的模等于 u → , v → \\overrightarrow\{\\mathbf\{u\}\}, \\overrightarrow\{\\mathbf\{v\}\} u,v构成的平行四边形的面积

#### 向量的并矢 ####

> 给定两个向量\{% raw %\}  x → = ( x 1 , x 2 , ⋯   , x n ) T , y → = ( y 1 , y 2 , ⋯   , y m ) T \\overrightarrow\{\\mathbf\{x\}\}=\\left(x\_\{1\}, x\_\{2\}, \\cdots, x\_\{n\}\\right)^\{T\}, \\overrightarrow\{\\mathbf\{y\}\}=\\left(y\_\{1\}, y\_\{2\}, \\cdots, y\_\{m\}\\right)^\{T\} x=(x1,x2,⋯,xn)T,y=(y1,y2,⋯,ym)T \{% endraw %\}，则向量的并矢记作：

x → y → = \[ x 1 y 1 x 1 y 2 ⋯ x 1 y m x 2 y 1 x 2 y 2 ⋯ x 2 y m ⋮ ⋮ ⋱ ⋮ x n y 1 x n y 2 ⋯ x n y m \] \\overrightarrow\{\\mathbf\{x\}\} \\overrightarrow\{\\mathbf\{y\}\}=\\left\[\\begin\{array\}\{cccc\} x\_\{1\} y\_\{1\} & x\_\{1\} y\_\{2\} & \\cdots & x\_\{1\} y\_\{m\} \\\\ x\_\{2\} y\_\{1\} & x\_\{2\} y\_\{2\} & \\cdots & x\_\{2\} y\_\{m\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ x\_\{n\} y\_\{1\} & x\_\{n\} y\_\{2\} & \\cdots & x\_\{n\} y\_\{m\} \\end\{array\}\\right\] xy=⎣⎢⎢⎢⎡x1y1x2y1⋮xny1x1y2x2y2⋮xny2⋯⋯⋱⋯x1ymx2ym⋮xnym⎦⎥⎥⎥⎤

> 也记作\{% raw %\} x → ⊗ y → \\overrightarrow\{\\mathbf\{x\}\} \\otimes \\overrightarrow\{\\mathbf\{y\}\} x⊗y\{% endraw %\}或者\{% raw %\} x → y → T \\overrightarrow\{\\mathbf\{x\}\} \\overrightarrow\{\\mathbf\{y\}\}^\{T\} xyT\{% endraw %\}。

### 矩阵运算 ###

> 给定两个矩阵\{% raw %\} A = ( a i , j ) ∈ R m × n , B = ( b i , j ) ∈ R m × n \\mathbf\{A\}=\\left(a\_\{i, j\}\\right) \\in \\mathbb\{R\}^\{m \\times n\}, \\mathbf\{B\}=\\left(b\_\{i, j\}\\right) \\in \\mathbb\{R\}^\{m \\times n\} A=(ai,j)∈Rm×n,B=(bi,j)∈Rm×n \{% endraw %\}，定义：

#### 阿达马积(Hadamard product)（又称作逐元素积） ####

A ∘ B = \[ a 1 , 1 b 1 , 1 a 1 , 2 b 1 , 2 ⋯ a 1 , n b 1 , n a 2 , 1 b 2 , 1 a 2 , 2 b 2 , 2 ⋯ a 2 , n b 2 , n ⋮ ⋮ ⋱ ⋮ a m , 1 b m , 1 a m , 2 b m , 2 ⋯ a m , n b m , n \] \\mathbf\{A\} \\circ \\mathbf\{B\}=\\left\[\\begin\{array\}\{cccc\} a\_\{1,1\} b\_\{1,1\} & a\_\{1,2\} b\_\{1,2\} & \\cdots & a\_\{1, n\} b\_\{1, n\} \\\\ a\_\{2,1\} b\_\{2,1\} & a\_\{2,2\} b\_\{2,2\} & \\cdots & a\_\{2, n\} b\_\{2, n\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a\_\{m, 1\} b\_\{m, 1\} & a\_\{m, 2\} b\_\{m, 2\} & \\cdots & a\_\{m, n\} b\_\{m, n\} \\end\{array\}\\right\] A∘B=⎣⎢⎢⎢⎡a1,1b1,1a2,1b2,1⋮am,1bm,1a1,2b1,2a2,2b2,2⋮am,2bm,2⋯⋯⋱⋯a1,nb1,na2,nb2,n⋮am,nbm,n⎦⎥⎥⎥⎤

#### 克罗内积(Kronnecker product) ####

A ⊗ B = \[ a 1 , 1 B a 1 , 2 B ⋯ a 1 , n B a 2 , 1 B a 2 , 2 B ⋯ a 2 , n B ⋮ ⋮ ⋱ ⋮ a m , 1 B a m , 2 B ⋯ a m , n B \] \\mathbf\{A\} \\otimes \\mathbf\{B\}=\\left\[\\begin\{array\}\{cccc\} a\_\{1,1\} \\mathbf\{B\} & a\_\{1,2\} \\mathbf\{B\} & \\cdots & a\_\{1, n\} \\mathbf\{B\} \\\\ a\_\{2,1\} \\mathbf\{B\} & a\_\{2,2\} \\mathbf\{B\} & \\cdots & a\_\{2, n\} \\mathbf\{B\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a\_\{m, 1\} \\mathbf\{B\} & a\_\{m, 2\} \\mathbf\{B\} & \\cdots & a\_\{m, n\} \\mathbf\{B\} \\end\{array\}\\right\] A⊗B=⎣⎢⎢⎢⎡a1,1Ba2,1B⋮am,1Ba1,2Ba2,2B⋮am,2B⋯⋯⋱⋯a1,nBa2,nB⋮am,nB⎦⎥⎥⎥⎤

### 偏导数 ###

*  标量对标量的偏导数： ∂ u ∂ v \\frac\{\\partial u\}\{\\partial v\} ∂v∂u 。
 *  标量对向量（  n n n 维向量）的偏导数 ：  ∂ u ∂ v → = ( ∂ u ∂ v 1 , ∂ u ∂ v 2 , ⋯   , ∂ u ∂ v n ) T \\frac\{\\partial u\}\{\\partial \\overrightarrow\{\\mathbf\{v\}\}\}=\\left(\\frac\{\\partial u\}\{\\partial v\_\{1\}\}, \\frac\{\\partial u\}\{\\partial v\_\{2\}\}, \\cdots, \\frac\{\\partial u\}\{\\partial v\_\{n\}\}\\right)^\{T\} ∂v∂u=(∂v1∂u,∂v2∂u,⋯,∂vn∂u)T。
 *  标量对矩阵(  m × n m \\times n m×n阶矩阵)的偏导数：

∂ u ∂ V = \[ ∂ u ∂ V 1 , 1 ∂ u ∂ V 1 , 2 ⋯ ∂ u ∂ V 1 , n ∂ u ∂ V 2 , 1 ∂ u ∂ V 2 , 2 ⋯ ∂ u ∂ V 2 , n ⋮ ⋮ ⋱ ⋮ ∂ u ∂ V m , 1 ∂ u ∂ V m , 2 ⋯ ∂ u ∂ V m , n \] \\frac\{\\partial u\}\{\\partial \\mathbf\{V\}\}=\\left\[\\begin\{array\}\{cccc\} \\frac\{\\partial u\}\{\\partial V\_\{1,1\}\} & \\frac\{\\partial u\}\{\\partial V\_\{1,2\}\} & \\cdots & \\frac\{\\partial u\}\{\\partial V\_\{1, n\}\} \\\\ \\frac\{\\partial u\}\{\\partial V\_\{2,1\}\} & \\frac\{\\partial u\}\{\\partial V\_\{2,2\}\} & \\cdots & \\frac\{\\partial u\}\{\\partial V\_\{2, n\}\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\frac\{\\partial u\}\{\\partial V\_\{m, 1\}\} & \\frac\{\\partial u\}\{\\partial V\_\{m, 2\}\} & \\cdots & \\frac\{\\partial u\}\{\\partial V\_\{m, n\}\} \\end\{array\}\\right\] ∂V∂u=⎣⎢⎢⎢⎢⎡∂V1,1∂u∂V2,1∂u⋮∂Vm,1∂u∂V1,2∂u∂V2,2∂u⋮∂Vm,2∂u⋯⋯⋱⋯∂V1,n∂u∂V2,n∂u⋮∂Vm,n∂u⎦⎥⎥⎥⎥⎤

*  向量（  m m m维向量）对标量的偏导数： ∂ u → ∂ v = ( ∂ u 1 ∂ v , ∂ u 2 ∂ v , ⋯   , ∂ u m ∂ v ) T \\frac\{\\partial \\overrightarrow\{\\mathbf\{u\}\}\}\{\\partial v\}=\\left(\\frac\{\\partial u\_\{1\}\}\{\\partial v\}, \\frac\{\\partial u\_\{2\}\}\{\\partial v\}, \\cdots, \\frac\{\\partial u\_\{m\}\}\{\\partial v\}\\right)^\{T\} ∂v∂u=(∂v∂u1,∂v∂u2,⋯,∂v∂um)T 。
 *  向量（  m m m维向量）对向量 (  n n n维向量) 的偏导数（雅可比矩阵，行优先）如果为列优先，则为矩阵的转置。

∂ u → ∂ v → = \[ ∂ u 1 ∂ v 1 ∂ u 1 ∂ v 2 ⋯ ∂ u 1 ∂ v n ∂ u 2 ∂ v 1 ∂ u 2 ∂ v 2 ⋯ ∂ u 2 ∂ v n ⋮ ⋮ ⋱ ⋮ ∂ u m ∂ v 1 ∂ u m ∂ v 2 ⋯ ∂ u m ∂ v n \] \\frac\{\\partial \\overrightarrow\{\\mathbf\{u\}\}\}\{\\partial \\overrightarrow\{\\mathbf\{v\}\}\}=\\left\[\\begin\{array\}\{cccc\} \\frac\{\\partial u\_\{1\}\}\{\\partial v\_\{1\}\} & \\frac\{\\partial u\_\{1\}\}\{\\partial v\_\{2\}\} & \\cdots & \\frac\{\\partial u\_\{1\}\}\{\\partial v\_\{n\}\} \\\\ \\frac\{\\partial u\_\{2\}\}\{\\partial v\_\{1\}\} & \\frac\{\\partial u\_\{2\}\}\{\\partial v\_\{2\}\} & \\cdots & \\frac\{\\partial u\_\{2\}\}\{\\partial v\_\{n\}\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\frac\{\\partial u\_\{m\}\}\{\\partial v\_\{1\}\} & \\frac\{\\partial u\_\{m\}\}\{\\partial v\_\{2\}\} & \\cdots & \\frac\{\\partial u\_\{m\}\}\{\\partial v\_\{n\}\} \\end\{array\}\\right\] ∂v∂u=⎣⎢⎢⎢⎡∂v1∂u1∂v1∂u2⋮∂v1∂um∂v2∂u1∂v2∂u2⋮∂v2∂um⋯⋯⋱⋯∂vn∂u1∂vn∂u2⋮∂vn∂um⎦⎥⎥⎥⎤

*  矩阵(  m × n m \\times n m×n阶矩阵)对标量的偏导数

∂ U ∂ v = \[ ∂ U 1 , 1 ∂ v ∂ U 1 , 2 ∂ v ⋯ ∂ U 1 , n ∂ v ∂ U 2 , 1 ∂ v ∂ U 2 , 2 ∂ v ⋯ ∂ U 2 , n ∂ v ⋮ ⋮ ⋱ ⋮ ∂ U m , 1 ∂ v ∂ U m , 2 ∂ v ⋯ ∂ U m , n ∂ v \] \\frac\{\\partial \\mathbf\{U\}\}\{\\partial v\}=\\left\[\\begin\{array\}\{cccc\} \\frac\{\\partial U\_\{1,1\}\}\{\\partial v\} & \\frac\{\\partial U\_\{1,2\}\}\{\\partial v\} & \\cdots & \\frac\{\\partial U\_\{1, n\}\}\{\\partial v\} \\\\ \\frac\{\\partial U\_\{2,1\}\}\{\\partial v\} & \\frac\{\\partial U\_\{2,2\}\}\{\\partial v\} & \\cdots & \\frac\{\\partial U\_\{2, n\}\}\{\\partial v\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\frac\{\\partial U\_\{m, 1\}\}\{\\partial v\} & \\frac\{\\partial U\_\{m, 2\}\}\{\\partial v\} & \\cdots & \\frac\{\\partial U\_\{m, n\}\}\{\\partial v\} \\end\{array\}\\right\] ∂v∂U=⎣⎢⎢⎢⎢⎡∂v∂U1,1∂v∂U2,1⋮∂v∂Um,1∂v∂U1,2∂v∂U2,2⋮∂v∂Um,2⋯⋯⋱⋯∂v∂U1,n∂v∂U2,n⋮∂v∂Um,n⎦⎥⎥⎥⎥⎤

### 参考资料 ###

*  http://www.huaxiaozhuan.com/%E6%95%B0%E5%AD%A6%E5%9F%BA%E7%A1%80/chapters/1\_algebra.html
 *  https://blog.csdn.net/zaishuiyifangxym/article/details/81673491