Matrix transformation

The basic matrix operations

Matrix addition

矩阵的加法满足下列运算律(A，B，C都是同型矩阵)：

A + B = B + A
(A + B) + C = A + (B + C)

应该注意的是只有同型矩阵之间才可以进行加法

Matrix subtraction

Matrix multiplication

Multiplication operations to meet the law

Matrix transpose

Transpose operation law satisfies

Conjugated

Conjugated matrix is defined: 
 . A conjugate complex matrix of 2 × 2 (constant real part, imaginary part neg)

Conjugate transpose

The conjugate transpose matrix is defined as: 
 can be written as: 
 or written 
 . A 2 × 2 conjugated complex transposed matrix as follows:

Matrix Multiplication

Multiplication of two matrices can be defined only if the number of rows equal to the number of columns of a first matrix A and matrix B of another. As A is an m × n matrix and B is an n × p matrix, and their product is a C m × p matrix 
 whose elements a

This product will be written as: 

Matrix multiplication law

Associative law: 

Left distributive law: 

Right distributive law: 

Matrix multiplication is not commutative.

Matrix transformation

Displacement

import matplotlib.pyplot as plt
import numpy as np

points = np.array([
    [0,0],
    [0,5],
    [3,5],
    [3,4],
    [1,4],
    [1,3],
    [2,3],
    [2,2],
    [1,2],
    [1,0],
    [0,0]
])

matrix = np.array([2,0])
newpoints = points + matrix

plt.plot(points[:,0],points[:,1])
plt.plot(newpoints[:,0],newpoints[:,1])
plt.xlim(-10,10)
plt.ylim(-10,10)
plt.show()

Rotation

import matplotlib.pyplot as plt
import numpy as np

points = np.array([
    [0,0],
    [0,5],
    [3,5],
    [3,4],
    [1,4],
    [1,3],
    [2,3],
    [2,2],
    [1,2],
    [1,0],
    [0,0]
])

matrix = np.array([
    [1,0],
    [0,-1]
])

newpoints = np.dot(points,matrix.T)
plt.plot(points[:,0],points[:,1])
plt.plot(newpoints[:,0],newpoints[:,1])
plt.xlim(-10,10)
plt.ylim(-10,10)
plt.show()

Scaling

import matplotlib.pyplot as plt
import numpy as np

points = np.array([
    [0,0],
    [0,5],
    [3,5],
    [3,4],
    [1,4],
    [1,3],
    [2,3],
    [2,2],
    [1,2],
    [1,0],
    [0,0]
])

matrix = np.array([
    [2,0],
    [0,1]
])

newpoints = np.dot(points,matrix.T)
plt.plot(points[:,0],points[:,1])
plt.plot(newpoints[:,0],newpoints[:,1])
plt.xlim(-10,10)
plt.ylim(-10,10)
plt.show()