如何找到矩形矩阵的列空间的基础? [英] How can I find a basis for the column space of a rectangular matrix?

问题描述

给出一个尺寸为m x n(其中n> m)的numpy ndarray，如何找到线性独立的列?

Given a numpy ndarray with dimensions m by n (where n>m), how can I find the linearly independent columns?

推荐答案

一种方法是使用

One way is to use the LU decomposition. The factor U will be of the same size as your matrix, but will be upper-triangular. In each row of U, pick the first nonzero element: these are pivot elements, which belong to linearly independent columns. A self-contained example:

import numpy as np
from scipy.linalg import lu
A = np.array([[1, 2, 3], [2, 4, 2]])     # example for testing 
U = lu(A)[2]
lin_indep_columns = [np.flatnonzero(U[i, :])[0] for i in range(U.shape[0])]

输出:[0，2]，表示A的第0列和第2列构成其列空间的基础.

Output: [0, 2], which means the 0th and 2nd columns of A form a basis for its column space.

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