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\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}
⎣⎢⎡a11⋮am1⋯⋱⋯a1n⋮amn⎦⎥⎤⎣⎢⎡b11⋮bn1⋯⋱⋯b1p⋮bnp⎦⎥⎤=⎣⎢⎡c11⋮cm1⋯⋱⋯c1p⋮cmp⎦⎥⎤
矩阵相乘的5种视角,互相等价
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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
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通过矩阵和向量乘法
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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,1≤k≤n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
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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,1≤k≤p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
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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)
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By Blocks
Invertibility
左逆矩阵(Left Inverse):
A − 1 A = I A^{-1}A = I A−1A=I
右逆矩阵(Right Inverse)
A A − 1 = I AA^{-1}=I AA−1=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=d1∗d2∗d3...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=SΛS−1
Symmetric case:
A = Q Λ Q − 1 = Q Λ Q T A=Q\Lambda Q^{-1} = Q\Lambda Q^T A=QΛQ−1=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,A∈Rm×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是 m m m阶正交矩阵, V V V是 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 σ1≥σ2≥...≥σp≥0
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} A∈Rm×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是 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。