Python学习笔记：求转置矩阵

一、矩阵转置

1、概述

矩阵的转置是矩阵的一种运算，在矩阵的所有运算法则中占有重要地位。

2、定义

设$A$为$m\times n$阶矩阵（即$m$行$n$列），第$i$行$j$列的元素是$a_{ij}$，即$A=(a_{ij}{)}_{m\times n}$，把$m\times n$矩阵$A$的行换成同序数的列得到一个$n\times m$矩阵，此矩阵叫做$A$的转置矩阵，记做$A^T$或$A^{\prime }$。

3、实例

$A=\left[\begin{array}{ccccc}7& 8& 1& 9& -3\\ 10& 4& 9& 4& 4\\ 7& 2& 8& 1& 4\\ 21& 4& 7& -4& 3\end{array}\right]$

$A^T=\left[\begin{array}{cccc}7& 10& 7& 21\\ 8& 4& 2& 4\\ 1& 9& 8& 7\\ 9& 4& 1& -4\\ -3& 4& 4& 3\end{array}\right]$

4、运算规律

矩阵的转置和加减乘除一样，也是一种运算，且满足下列运算规律：

• $(A^T{)}^T=A$
• $(A+B{)}^T=A^T+B^T$
• $(kA{)}^T=k{A}^T$
• $(AB{)}^T=B^T{A}^T$

5、对阵矩阵

（1）定义

设$A$为$n$阶矩阵，如果满足$A^T=A$，即$a_{ij}=a_{ji}(i,j=1,2,3,...,n)$，那么$A$称为对称矩阵。

（2）实例

$A=\left[\begin{array}{cccc}7& 8& 1& 6\\ 8& 4& 9& 4\\ 1& 9& 8& 7\\ 6& 4& 7& 2\end{array}\right]$

二、编程求转置矩阵

1、编写程序 - 矩阵转置.py

# -*- coding: utf-8 -*-
"""
Created on Sun Oct 11 21:05:53 2020

@author: howard

实现矩阵转置
"""

matrix = [
     [7, 8, 1, 9, -3],
     [10, 4, 9, 4, 4],
     [7, 2, 8, 1, 4],
     [21, 4, 7, -4, 3]
    ]

print('原矩阵：')
for i in range(len(matrix)):
    for j in range(len(matrix[0])):
        print('{0:-2d}'.format(matrix[i][j]), end='\t')
    print()
    
# 求转置矩阵
transposed = []
for j in range(len(matrix[0])):
    transposed_row = []
    for row in matrix:
        transposed_row.append(row[j])
    transposed.append(transposed_row)
  
print('转置矩阵：')
for i in range(len(transposed)):
    for j in range(len(transposed[0])):
        print('{0:-2d}'.format(transposed[i][j]), end='\t')
    print()

2、运行程序，查看结果

3、优化代码 - 定义求转置矩阵的函数

# -*- coding: utf-8 -*-
"""
Created on Sun Oct 11 21:05:53 2020

@author: howard

实现矩阵转置
"""

def transpose(matrix):
    transposed = []
    for j in range(len(matrix[0])):
        transposed_row = []
        for row in matrix:
            transposed_row.append(row[j])
        transposed.append(transposed_row)
    return transposed

matrix = [
     [7, 8, 1, 9, -3],
     [10, 4, 9, 4, 4],
     [7, 2, 8, 1, 4],
     [21, 4, 7, -4, 3]
    ]

print('原矩阵：')
for i in range(len(matrix)):
    for j in range(len(matrix[0])):
        print('{0:-2d}'.format(matrix[i][j]), end='\t')
    print()
    
# 求转置矩阵
transposed = transpose(matrix)
  
print('转置矩阵：')
for i in range(len(transposed)):
    for j in range(len(transposed[0])):
        print('{0:-2d}'.format(transposed[i][j]), end='\t')
    print()

4、运行程序，查看效果

三、利用NumPy数组实现矩阵转置

1、编写程序 - 实现矩阵转置.py

2、运行程序，查看结果