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Linear Algebra (一)

ZSACH 2022-03-30 阅读 52

Multiply

A B = C AB = C AB=C
[ a 11 ⋯ a 1 n ⋮ ⋱ ⋮ a m 1 ⋯ a m n ] [ b 11 ⋯ b 1 p ⋮ ⋱ ⋮ b n 1 ⋯ b n p ] = [ c 11 ⋯ c 1 p ⋮ ⋱ ⋮ c m 1 ⋯ c m p ] \begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn}\end{bmatrix}\begin{bmatrix} b_{11} & \cdots & b_{1p} \\ \vdots & \ddots & \vdots \\ b_{n1} & \cdots & b_{np}\end{bmatrix}=\begin{bmatrix} c_{11} & \cdots & c_{1p} \\ \vdots & \ddots & \vdots \\ c_{m1} & \cdots & c_{mp}\end{bmatrix} a11am1a1namnb11bn1b1pbnp=c11cm1c1pcmp

矩阵相乘的5种视角,互相等价

  • 常规视角

    • c i j = ∑ k = 1 n a i k b k j c_{ij} = \sum_{k=1}^n a_{ik}b_{kj} cij=k=1naikbkj
  • 通过矩阵和向量乘法

    • 右乘:C 的每个列向量 c j \bold{c}_j cj由A的列向量 a k , 1 ≤ k ≤ n \bold{a}_k,1\le k\le n ak,1kn的线性组合构成 c j = ∑ k = 1 n b k j a k \bold{c}_j = \sum_{k=1}^n b_{kj} \bold{a}_k cj=k=1nbkjak

    • 左乘:C 的没个行向量 c i \bold{c}_i ci 由B的行向量 b k , 1 ≤ k ≤ p \bold{b}_k,1\le k\le p bk,1kp的线性组合构成 c i = ∑ k = 1 p a i k b k \bold{c}_i=\sum_{k=1}^pa_{ik}\bold{b}_k ci=k=1paikbk

  • A B = ∑ i ( c o l u m n i O f A ) ( r o w i O f B ) AB = \sum_i (column_iOf A)(row_iOf B) AB=i(columniOfA)(rowiOfB)

  • By Blocks

Invertibility

左逆矩阵(Left Inverse):

A − 1 A = I A^{-1}A = I A1A=I

右逆矩阵(Right Inverse)

A A − 1 = I AA^{-1}=I AA1=I

Dependence:

向量 V 1 , V 2 , . . , V n V_1,V_2,..,V_n V1,V2,..,Vn,存在组合非全零实数 c 1 , c 2 , . . . , c n c_1, c_2,...,c_n c1,c2,...,cn,满足 ∑ i = 1 n c i V i = 0 \sum_{i=1}^nc_iV_i = 0 i=1nciVi=0,则称向量线性相关。

行列式

矩阵行列式的三个性质:

  • d e t ( I ) = 1 det(I) = 1 det(I)=1
  • Exchange rows , reverse sign of determinant
  • Linear for each row

由此三个性质推导出行列式的以下性质:

  • two equal rows => det = 0
  • subtract l × r o w i l\times row_i l×rowi from r o w j row_j rowj, det does not change.
  • Row of zeros => det = 0
  • Triangle matrix => d e t = d 1 ∗ d 2 ∗ d 3... d n det = d1*d2*d3...dn det=d1d2d3...dn
  • det = 0 exactly when A is singular
  • d e t ( A B ) = d e t ( A ) d e t ( B ) det(AB) = det(A)det(B) det(AB)=det(A)det(B)
  • d e t A T = d e t A detA^T = det A detAT=detA

行列式的定义

d e t ( A ) = ∑ n ! a 1 α a 2 β a 1 γ . . . a n ω det(A) = \sum_{n!} a_{1\alpha}a_{2\beta}a_{1\gamma}...a_{n\omega} det(A)=n!a1αa2βa1γ...anω

( α , β , γ , . . . , ω ) (\alpha,\beta,\gamma,...,\omega) (α,β,γ,...,ω)is permutation of (1,2,3,…,n)

Singular

Exist a none zero vector X, satisfiy A x = 0 Ax = 0 Ax=0,then A is singular.

特征值和特征向量

定义:A是n阶矩阵,若实数 λ \lambda λ和n维非零向量 α \alpha α满足 A α = λ α A\alpha = \lambda\alpha Aα=λα,则称 λ \lambda λ为A的特征值, α \alpha α为A的特征向量。

  • If A is singular, the λ = 0 \lambda=0 λ=0 is an eigenvalue.
  • Trace: ∑ i λ i = ∑ a i i \sum_i \lambda_i = \sum a_{ii} iλi=aii
  • Determinant : d e t = ∏ i λ i det = \prod_i \lambda_i det=iλi
  • 对称或近似对称,特征值是实数,否则可能是复数。

对称矩阵

对于对称举矩阵

  • the eigenvalues are also Real
  • the eigenvectors are Perpendicular

Usual case:

A = S Λ S − 1 A = S\Lambda S^{-1} A=SΛS1

Symmetric case:

A = Q Λ Q − 1 = Q Λ Q T A=Q\Lambda Q^{-1} = Q\Lambda Q^T A=QΛQ1=QΛQT

奇异值分解

定义:矩阵的奇异值分解是指,将一个非零的 m × n m\times n m×n实矩阵 A , A ∈ R m × n A,A\in R^{m\times n} A,ARm×n,表示为以下三个实矩阵乘积形式的运算,即进行矩阵的因子分解:

A = U Σ V T A= U\Sigma V^T A=UΣVT

其中 U U U m m m阶正交矩阵, V V V n n n阶正交矩阵, Σ \Sigma Σ是由降序排列的非负的对角线元素组成的 m × n m\times n m×n矩形对角阵,满足

U U T = I V V T = I Σ = d i a g ( σ 1 , σ 2 , . . . , σ p ) UU^T=I\\VV^T=I\\ \Sigma=diag(\sigma_1,\sigma_2,...,\sigma_p) UUT=IVVT=IΣ=diag(σ1,σ2,...,σp)

σ 1 ≥ σ 2 ≥ . . . ≥ σ p ≥ 0 \sigma_1\ge \sigma_2\ge ...\ge\sigma_p\ge 0 σ1σ2...σp0

p = m i n ( m , n ) p=min(m,n) p=min(m,n)

U Σ V T U\Sigma V^T UΣVT称为矩阵A的奇异值分解(singular value decomposition), σ i \sigma_i σi称为矩阵A的奇异值(singular value), U U U的列向量称为左奇异向量,V的列向量称为右奇异向量。

奇异值分解定理:若A为 m × n m\times n m×n实矩阵, A ∈ R m × n A\in R^{m\times n} ARm×n,则A的奇异值分解存在

A = U Σ V T A=U\Sigma V^T A=UΣVT

其中 U U U是m阶正交矩阵, V V V是n阶正交矩阵, Σ \Sigma Σ m × n m\times n m×n矩形对角矩阵,其对角线元素非负,且降序排列。

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