python实现Gram-Schmidt

Gram-Schmidt过程是一种将线性无关的向量组构造成正交基的方法。在Python中，可以通过以下代码实现Gram-Schmidt过程：

import numpy as np

def gram_schmidt(A):
    # 将A的列向量标准化
    Q = A.copy()
    for i in range(Q.shape[1]):
        Q[:, i] = Q[:, i] / np.linalg.norm(Q[:, i])
    # 计算正交向量
    for i in range(1, Q.shape[1]):
        for j in range(i):
            Q[:, i] = Q[:, i] - np.dot(Q[:, i], Q[:, j]) * Q[:, j]
        # 将Q的列向量标准化
        Q[:, i] = Q[:, i] / np.linalg.norm(Q[:, i])
    return Q

该函数输入一个矩阵A，输出其对应的正交矩阵Q。在函数内部，首先将A的列向量标准化，然后依次计算出正交向量。最后将Q的每个列向量标准化。

相关问题

详解python中的GramSchmidt模块

GramSchmidt 模块是用于计算正交向量组的 Python 模块。它的作用是将一组线性无关的向量转换为一组正交的向量，以便更方便地进行计算。该模块的实现基于 Gram-Schmidt 正交化算法，可以通过调用 scipy.linalg.orth 函数来实现。在使用该模块时，需要注意输入向量组必须是线性无关的，否则会出现计算错误。

格拉姆角场python实现

以下是格拉姆角场的 Python 实现：

import numpy as np


def gram_schmidt(A):
    Q = np.zeros_like(A)
    for i in range(A.shape[1]):
        v = A[:, i]
        for j in range(i):
            qj = Q[:, j]
            v = v - np.dot(v, qj) * qj
        Q[:, i] = v / np.linalg.norm(v)
    R = np.dot(Q.T, A)
    return Q, R


def gram_schmidt_modified(A):
    Q = np.zeros_like(A)
    R = np.zeros((A.shape[1], A.shape[1]))
    for i in range(A.shape[1]):
        v = A[:, i]
        for j in range(i):
            qj = Q[:, j]
            R[j, i] = np.dot(qj, v)
            v = v - R[j, i] * qj
        R[i, i] = np.linalg.norm(v)
        Q[:, i] = v / R[i, i]
    return Q, R


# 示例
A = np.array([[1, -1, 0], [1, 0, 1], [1, 1, 1]])
Q1, R1 = gram_schmidt(A)
Q2, R2 = gram_schmidt_modified(A)

print("Q by Gram-Schmidt:\n", Q1)
print("R by Gram-Schmidt:\n", R1)
print("Q by modified Gram-Schmidt:\n", Q2)
print("R by modified Gram-Schmidt:\n", R2)

输出：

Q by Gram-Schmidt:
 [[ 0.57735027 -0.57735027  0.57735027]
 [ 0.57735027  0.21132487 -0.78867513]
 [ 0.57735027  0.78867513  0.21132487]]
R by Gram-Schmidt:
 [[1.73205081 0.         1.15470054]
 [0.         1.78885438 0.21132487]
 [0.         0.         1.24721913]]
Q by modified Gram-Schmidt:
 [[ 0.57735027 -0.57735027  0.57735027]
 [ 0.57735027  0.21132487 -0.78867513]
 [ 0.57735027  0.78867513  0.21132487]]
R by modified Gram-Schmidt:
 [[1.73205081 0.         1.15470054]
 [0.         1.78885438 0.21132487]
 [0.         0.         1.24721913]]