norm

Vector and matrix norms

Syntax

n = norm(v)

n = norm(v,p)

n = norm(X)

n = norm(X,p)

n = norm(X,'fro')

Description

n = norm（v）返回向量v的歐幾里德範數。該範數也稱為2範數，向量幅度或歐幾里德長度。

n = norm（v，p）返回廣義向量p範數。

n = norm（X）返回矩陣X的2範數或最大奇異值，其近似為max（svd（X））。

n = norm（X，p）返回矩陣X的p範數，其中p為1,2或Inf：

• 如果p = 1，則n是矩陣的最大絕對列和。

• 如果p = 2，則n近似為max（svd（X））。 這相當於norm（X）。

• 如果p = Inf，那麼n是矩陣的最大絕對行和。

n = norm（X，'fro'）返回矩陣X的Frobenius範數。

有關範數的基礎知識，見博文：【 MATLAB 】範數的必備基礎知識

下面舉例說明：

Vector Magnitude（向量幅度）

%Create a vector and calculate the magnitude.

v = [1 -2 3];
n = norm(v)
% n = 3.7417

1-Norm of Vector

clc
clear
close all

% Calculate the 1-norm of a vector, which is the sum of the element magnitudes.

X = [-2 3 -1];
n = norm(X,1)
% n = 6
 

Euclidean Distance Between Two Points

clc
clear
close all

% Calculate the distance between two points as the norm of the difference between the vector elements.
% 
% Create two vectors representing the (x,y) coordinates for two points on the Euclidean plane.

a = [0 3];
b = [-2 1];
% Use norm to calculate the distance between the points.

d = norm(b-a)
 

d =

    2.8284
幾何上，兩點之間的距離：

2-Norm of Matrix

clc
clear
close all

% Calculate the 2-norm of a matrix, which is the largest singular value.

X = [2 0 1;-1 1 0;-3 3 0];
n = norm(X)
% n = 4.7234

Frobenius Norm of Sparse Matrix

clc
clear
close all

% 使用'fro'計算稀疏矩陣的Frobenius範數，該範數計算列向量的2範數S（:)。

S = sparse(1:25,1:25,1);
n = norm(S,'fro')
% n = 5