Section01 Vectors and Linear

$$\boldsymbol{Ax} = \boldsymbol{b}\\ (\boldsymbol{x} = \left[ \begin{matrix} x_{1} & x_{2} & \cdots & x_{n} \end{matrix} \right]^{\mathbf{T}})$$

Perspective from Rows

Two or more hyperplane($\mathbb{R}^{n-1}$) meeting at a single point. (when $A$ is Invertible

Perspective from Columns

Linear Combination of the columns.

• Coefficient Matrix $$\boldsymbol{A} = \left[ \begin{matrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{matrix} \right]$$

• Matrix equation $$\boldsymbol{Ax} = \boldsymbol{b} \\\\ \left[ \begin{matrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{matrix} \right] \left[ \begin{matrix} x_{1} \\ x_{2} \\ \vdots \\ x_{n} \end{matrix} \right] = \left[ \begin{matrix} b_{1} \\ b_{2} \\ \vdots \\ b_{m} \end{matrix} \right]$$

• Identity Matrix
  $$\boldsymbol{I} = \left[ \begin{matrix} 1 & 0 & \cdots & 0\\ 0 & 1 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1\\ \end{matrix} \right]$$

  • for any Matrix $\boldsymbol{A}$ $$\boldsymbol{IA} = \boldsymbol{A}$$

Multiplication in Julia

• create Matrix and Vector

  # Create a matrix
  A = [1 2 3; 2 5 2; 6 -3 1];
  
  # Create a vector
  x = [0 0 2]';
  # or
  x = [0,0,2];
  

• Matrix Multiplication

  b = A * x
  

  • perspective of rows

    b = [A[1,:]' * x, A[2,:]' * x, A[3,:]' * x]
    

  • perspective of columns

    b = x[1] * A[:,1] + x[2] * A[:,2] + x[3] * A[:,3]
    

Markov Matrix

• example $$\begin{array}{c} \left\{ \begin{array}{ll} \boldsymbol{M} = \left[ \begin{matrix} 0.8 & 0.3 \\ 0.2 & 0.7 \end{matrix} \right] \\ \boldsymbol{\mu}_{1} = \boldsymbol{M} \left[ \begin{matrix} 1 \\ 0 \end{matrix} \right] \end{array} \right.\\\\ \boldsymbol{\mu}_{k} = \boldsymbol{M\mu}_{k-1}\\ \text{sum(}\boldsymbol{\mu}_{k}\text{)} = 1 \end{array}$$

Magic Matrix

• example $$\begin{array}{c} \boldsymbol{A} \text{ is a n-dimenssion square matrix, is consist of }\{ 1,2,\cdots,n^{2} - 1 ,n^{2} \}\\\text{ the sums of its columns, rows and diagonals is always equal to } \frac{\sum_{i=1}^{n^{2}}i}{n} \end{array}$$

• generating in Julia

  # need to add the MatrixDepot Packages
  using MatrixDepot
  M = matrixdepot("magic", Int, 5);