四元数奇异值分解 Python代码

以下是使用Python实现四元数奇异值分解（Quaternion Singular Value Decomposition）的代码：

import numpy as np
from scipy.linalg import sqrtm

def quaternion_svd(Q):
    """
    Perform Quaternion Singular Value Decomposition (QSVD) on a quaternion matrix Q.
    
    Args:
        Q (np.array): A 4x4 quaternion matrix.
    
    Returns:
        U (np.array): A 4x4 unitary quaternion matrix.
        sigma (np.array): A 4x1 vector of singular values.
        V (np.array): A 4x4 unitary quaternion matrix.
    """
    # Construct the Hermitian matrix H from Q
    H = np.zeros((4, 4), dtype=np.complex64)
    H[:3, :3] = Q[1:, 1:] - Q[0, 0] * np.identity(3)
    H[:3, 3] = Q[1:, 0]
    H[3, :3] = np.conj(Q[1:, 0])
    H[3, 3] = Q[0, 0]
    
    # Compute the eigenvectors and eigenvalues of H
    eigvals, eigvecs = np.linalg.eig(H)
    
    # Sort the eigenvectors based on the magnitude of their eigenvalues
    idx = np.argsort(np.abs(eigvals))[::-1]
    eigvecs = eigvecs[:, idx]
    
    # Compute the left and right singular vectors of Q
    U = np.zeros((4, 4), dtype=np.complex64)
    V = np.zeros((4, 4), dtype=np.complex64)
    for i in range(4):
        v = eigvecs[:, i]
        u = Q @ v
        U[:, i] = u / np.linalg.norm(u)
        V[:, i] = v / np.linalg.norm(v)
    
    # Compute the singular values of Q
    sigma = np.sqrt(np.abs(eigvals[idx]))
    
    return U, sigma, V

这个函数接受一个 $4 \times 4$ 的四元数矩阵作为输入，返回左奇异向量矩阵 $U$，奇异值向量 $\sigma$ 和右奇异向量矩阵 $V$。其中，左奇异向量矩阵和右奇异向量矩阵都是 $4 \times 4$ 的 unitary quaternion 矩阵，而奇异值向量是一个 $4 \times 1$ 的向量。

注意，这个代码依赖于 NumPy 和 SciPy 库。