2.1 Linear Algebra

2.1.1 Basic Concepts

• Scalar: A number
• Vector: A column of ordered numbers $$ x=\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\x_n \end{bmatrix}$$
  • norm of vector: the length
    • $L_p$ norm: $||x||_p = (\sum_{i=1}^{n}|x_i|^p)^{1/p}$
    • $L_1$ norm: $||x||_1 = (\sum_{i=1}^{n}|x_i|)$
    • $L_2$ norm: $||x||_2 = \sqrt{\sum_{i=1}^{n}{x_i}^2}$ --> used to measure the length of vectors
    • $L_\infty$ norm: $||x||_\infty = max_i |x_i|$
  • the distance of two vectors $x_1, x_2$ is $$D_p(x_1, x_2) = ||x_1 - x_2||_p$$
  • a set of vectors $x_1, x_2, \cdots, x_m$ are linearly independent iff there does not exist a set of scalars $\lambda_1, \lambda_2, \cdots, \lambda_m$, which are not all 0, such that $$\lambda_1 x_1 + \lambda_2 x_2 + \cdots + \lambda_m x_m = 0$$
• Matrix: 2-dimensional array $$ A=\begin{bmatrix} a_{11} &a_{12}& \cdots& a_{1n}  \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots &\vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$$ where $A\in \mathbb{R}^{m\times n}$ 
• Matrix Product: For $A\in \mathbb{R}^{m\times n} $ and $B\in \mathbb{R}^{n\times p}$, matrix product of AB is denoted as $C\in \mathbb{R}^{m\times p}$ and calculated as $$C_{ij} = \sum_{k=1}^{n}A_{ik}B_{kj} $$
  • Features
    • (AB)C = A(BC) --> always true
    • AB = BA --> not always true
• Determinant: $det(A) = \sum_{k_1 k_2 \cdots k_n} (-1)^{\tau(k_1k_2 \cdots k_n)}a_{1k_1}a_{2k_2} \cdots a_{nk_n} $ 
• Inversion Matrix: If matrix A is a square matrix, the inverse matrix of A satisfies $A^{-1}A=I$ 
• Transpose: $A_{ij}^T = A_{ji}$
• Hadamard product: $C_{ij}=A_{ij}B_{ij}$ where  $A\in \mathbb{R}^{m\times n}$, $B\in \mathbb{R}^{m\times n}$ and $C\in \mathbb{R}^{m\times n}$
  
  • en.wikipedia.org/wiki/Hadamard_product_(matrices)

Hadamard product (matrices) - Wikipedia

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2.1.2 Eigendecomposition (고유값 분해)

• Let A be a matrix in $\mathbb{R}^{n\times n}$, $v\in \mathbb{C}^n$ and $\lambda \in \mathbb{C}$  $$Av=\lambda v$$
  • $\lambda$ : eigenvalue of A
  • v: eigenvector of A

수식으로 표현한 eigenvector & eigenvalue

• Let $V=[v_1 v_2 \cdots v_n] $ (V: invertible matrix)
  • Eigendecomposition of A(diagonalization) $$ A=Vdiag(\lambda V^{-1}) $$

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2.1.3 Singular Value Decomposition(특이값 분해 SVD)

eigendecomposition(고유값 분해)와 같이 행렬을 대각화하는 방법

• singular value
  • r: the rank of $A^T A$
  • $ 0 < \sigma_r \leq \cdots \leq \sigma_2 \leq \sigma_1$ such that $1 \leq i \leq r$
  • $v_i$ is an eigen vector of $A^T A$ with corresponding eigenvalue $sigma_i^2$.
  • singular value of A is $\sigma_1, \sigma_2,\cdots ,\sigma_r$.
• singular value decomposition  $$ A = U\sum V^T$$
  • $U\in \mathbb{R}^{m\times m}$, $V\in \mathbb{R}^{n \times n}$ and $\sum$  (where $\sum_{ij} = \sigma_i$ (iff $i=j \leq r $))