吴恩达机器学习笔记（三）–线性代数回顾

学习基于：吴恩达机器学习.

1. Matrix and Vector

• Matrix is a rectangular array of numbers
• The dimension of matrix is the number of rows and columns of a matrix
• Matrix elements are entries of matrix
• $A_{ij}$Aij​ refers to the entry in the $i^{th}$ith row and $j^{th}$jth column
• Vector is an n x 1 matrix
  
  
  

2. Addition and Scalar Multiplication

• Addition and subtraction are element-wise, so you simply add or subtract each corresponding element:
  $\left[ \begin{matrix} a & c \\ b & d \end{matrix} \right] + \left[ \begin{matrix} w & y \\ x & z \end{matrix} \right] = \left[ \begin{matrix} a+w & c+y \\ b+x & d+z \end{matrix} \right]$[ab​cd​]+[wx​yz​]=[a+wb+x​c+yd+z​]
• Subtracting Matrices:
  $\left[ \begin{matrix} a & c \\ b & d \end{matrix} \right] - \left[ \begin{matrix} w & y \\ x & z \end{matrix} \right] = \left[ \begin{matrix} a-w & c-y \\ b-x & d-z \end{matrix} \right]$[ab​cd​]−[wx​yz​]=[a−wb−x​c−yd−z​]

To add or subtract two matrices, their dimensions must be the same.

• In scalar multiplication, we simply multiply every element by the scalar value:
  $\left[ \begin{matrix} a & c \\ b & d \end{matrix} \right] \times \lambda = \left[ \begin{matrix} \lambda a & \lambda c \\ \lambda b & \lambda d \end{matrix} \right]$[ab​cd​]×λ=[λaλb​λcλd​]

3. Matrix-Matrix Multiplication

1) Matrix-Vector Multiplication

We map the column of the vector onto each row of the matrix, multiplying each element and summing the result.
$\left[ \begin{matrix} a & c \\ b & d \end{matrix} \right] \times \left[ \begin{matrix} x \\ y \\ \end{matrix} \right]= \left[ \begin{matrix} ax+cy \\ bx+dy \end{matrix} \right]$[ab​cd​]×[xy​]=[ax+cybx+dy​]
An m x n matrix multiplied by an n x 1 vector results in an m x 1 vector.

2) Matrix-Matrix Multiplication

We multiply two matrices by breaking it into several vector multiplications and concatenating the result.
$\left[ \begin{matrix} a & c \\ b & d \end{matrix} \right] \times \left[ \begin{matrix} w & y \\ x & z \end{matrix} \right]= \left[ \begin{matrix} aw+cx & ay+cz \\ bw+dx & by+dz \end{matrix} \right]$[ab​cd​]×[wx​yz​]=[aw+cxbw+dx​ay+czby+dz​]

4. Matrix Multiplication Properties

• Matrices are not commutative:
  • A$\times$×B ≠ B$\times$×A 　　　　　 (Except B is an unit matrix)
• Matrices are associative:
  • (A$\times$×B)$\times$×C=A$\times$×(B$\times$×C)

5. Inverse and Transpose

• The inverse of a matrix A is denoted $A^{-1}$A−1. Multiplying by the inverse results in the identity matrix.
  • $AA^{-1} = A^{-1}A = I$AA−1=A−1A=I
• The transposition of a matrix is like rotating the matrix 90° in clockwise direction and then reversing it.
  • $B_{ij} = A_{ji}$Bij​=Aji​