Scalar, Vector and Matrix

Scalar: 하나의 숫자

$s \in \mathbb{R}$ e. g) 3, 8

Vector: 정렬된 숫자 리스트

$S = \begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_n\end{bmatrix} \in \mathbb{R}^n$

Matrix: 2차원을 지닌 숫자 배열

Matrix size: 3 X 2 3 column, 2 row
Column vector: a vertical vector
Row vector: a horizontal vector

$S = \begin{bmatrix}1 & 2 \\ 3 & 4 \\ 5 & 6\end{bmatrix} \in \mathbb{R}^{3\times2}$

Matrix Notation

Square matrix (#rows = #columns)

$A = \begin{bmatrix}1 &2 \\ 3 & 4\end{bmatrix}$

Rectangular matrix (#rows != #columns)

$A = \begin{bmatrix}1 &2 \\ 3 & 4 \\ 5 & 6\end{bmatrix}$

Transpose of matrix (중앙 대각행렬 기준으로 교차)

$A = \begin{bmatrix}1 &2 \\ 3 & 4 \\ 5 & 6\end{bmatrix}, A^T = \begin{bmatrix}1 & 3 & 5 \\ 2 & 4 & 6\end{bmatrix}$

$A_{i,j}$ : (i, j)-th component of A, e.g., $A_{2,1} = 3$

$A_{i,:}$ : i-th component of A, e.g., $A_{2,:} = \begin{bmatrix}3 & 4\end{bmatrix}$

$A_{:,j}$ : j-th component of A, e.g., $A_{:,2} = \begin{bmatrix}2 \\ 4 \\ 6\end{bmatrix}$

Vector/Matrix Additions and Multiplications

Additions

$C_{ij} = A_{ij} + B{ij} \\ A,B,C \in \mathbb{R}^{m \times n}$

A,B,C는 같은 크기를 가져야 한다.

Scalar multiple of vector/matrx

$2\begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix} = \begin{bmatrix}2 \\ 4 \\ 6\end{bmatrix}, 2\begin{bmatrix}1 & 2\\ 3 & 4\\ 5 & 6\end{bmatrix} = \begin{bmatrix}2 & 4\\ 6 & 8\\ 10 & 12\end{bmatrix}$