矩阵范数及python代码

1. 矩阵范数

假设我们有一个矩阵A 表示如下：
$$\begin{array}{c}A=\left[\begin{array}{ccc}1& 2& 3\\ & & \\ 4& 5& 6\\ & & \\ 7& 8& 9\end{array}\right]\end{array}$$

• L1范数表示A的列范数，即将矩阵A的按列划分后，求出最大的列
  $$\begin{array}{c}||A|{|}_1=\underset{1\le j\le n}{\max }\sum _{i=1}^n|a_{ij}|\end{array}$$
  $$\begin{array}{c}||A|{|}_1=\max [12,15,18]=18\end{array}$$
• $L_{\infty }$范数表示A的行范数，即将矩阵A的按行划分后，求出最大的行
  $$\begin{array}{c}||A|{|}_1=\underset{1\le i\le n}{\max }\sum _{i=1}^n|a_{ij}|\end{array}$$
  $$\begin{array}{c}||A|{|}_{\infty }=\max [6,15,24]=24\end{array}$$
• L2范数表示$A^{T}A$最大特征值的根号值,先求$A^{T}A$,再求最大特征值的根号
  $$\begin{array}{c}||A|{|}_2=\sqrt{{\lambda }_{\max }(A^{T}A)}=16.848\end{array}$$

2. 代码：

#!/usr/bin/env python
# -*- coding:utf-8 -*-
# @FileName  :matrix_normtest.py
# @Time      :2024/9/26 9:53
# @Author    :Jason Zhang
import numpy as np

np.set_printoptions(suppress=True, precision=3)
np.random.seed(20230512)


class Eigenvalue():
    def __init__(self, array):
        self.array = array
        self.value, self.vector = np.linalg.eig(self.array)
        self.L1 = np.linalg.norm(self.array, 1)
        self.L2 = np.linalg.norm(self.array, 2)
        self.Linfty = np.linalg.norm(self.array, np.inf)

    def __str__(self):
        return f"array:\n{self.array}:\n eigenvalue:\n{self.value}\n,eigenveotr:\n{self.vector}" \
               f"\nL1:={self.L1},\nL2:={self.L2}\nL_inf:={self.Linfty}"


if __name__ == "__main__":
    run_code = 0
    A = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
    ATA = np.dot(A.T, A)
    A_eigen = Eigenvalue(A)
    print(A_eigen)
    ATA_eigen = Eigenvalue(ATA)
    print(ATA_eigen)

• 运行结果：

array:
[[1 2 3]
 [4 5 6]
 [7 8 9]]:
 eigenvalue:
[16.117 -1.117 -0.   ]
,eigenveotr:
[[-0.232 -0.786  0.408]
 [-0.525 -0.087 -0.816]
 [-0.819  0.612  0.408]]
L1:=18.0,
L2:=16.84810335261421
L_inf:=24.0
array:
[[ 66  78  90]
 [ 78  93 108]
 [ 90 108 126]]:
 eigenvalue:
[283.859   1.141   0.   ]
,eigenveotr:
[[-0.48  -0.777  0.408]
 [-0.572 -0.076 -0.816]
 [-0.665  0.625  0.408]]
L1:=324.0,
L2:=283.8585865803701
L_inf:=324.0