用python编写关于这个SVD分解问题的解决代码:Read data set A.cav as a matrix A e Rx⁵.Compaute the SVD of A and rport (a) the fourth singular value, and (b) the rank of A? Compute the cigendecomposition of A. (c)For every non-zero cigenvalhe, report it and its associated cigenvector. How many non-zero eigrnvalues are there? Compute A, for k=3. (d)What is [A-Ail}? (e)What is |A-A? Ceater A. Run PCA to find the best 3-dimensional subspace F to minimize [A-mp(4)Report (0 |A-πp(4)} and (g)|A-π(A)
时间: 2024-03-06 15:52:15 浏览: 30
下面是Python中使用Numpy库解决该问题的示例代码:
```python
import numpy as np
# 读取数据集A.cav
A = np.loadtxt('A.cav')
# 计算A的SVD
U, S, VT = np.linalg.svd(A)
# (a) 输出第四个奇异值
print("The fourth singular value of A is:", S[3])
# (b) 输出A的秩
rank_A = np.linalg.matrix_rank(A)
print("The rank of A is:", rank_A)
# (c) 计算特征分解
eigvals, eigvecs = np.linalg.eig(np.dot(A.T, A))
# 输出非零特征值及其对应特征向量
for i in range(len(eigvals)):
if eigvals[i] != 0:
print("The %d-th non-zero eigenvalue is %f, and its associated eigenvector is:" % (i+1, eigvals[i]), eigvecs[:, i])
# 统计非零特征值的个数
nonzero_eigvals = np.count_nonzero(eigvals)
print("There are %d non-zero eigenvalues." % nonzero_eigvals)
# (d) 计算A_k
k = 3
Ak = np.dot(np.dot(U[:, :k], np.diag(S[:k])), VT[:k, :])
print("A_k for k=3 is:\n", Ak)
# (e) 计算A-Ak
A_Ak = A - Ak
print("A - A_k is:\n", A_Ak)
# (f) PCA降维
m = 4
mean_A = np.mean(A, axis=0)
A_centered = A - mean_A
C = np.dot(A_centered.T, A_centered)
eigvals_pca, eigvecs_pca = np.linalg.eig(C)
idx = eigvals_pca.argsort()[::-1]
eigvals_pca = eigvals_pca[idx]
eigvecs_pca = eigvecs_pca[:, idx]
F = np.dot(A_centered, eigvecs_pca[:, :3])
print("The best 3-dimensional subspace F is:\n", F)
# (g) 计算|A-Ak|和|A-π(A)|
norm_A_Ak = np.linalg.norm(A_Ak)
print("|A - A_k| is:", norm_A_Ak)
norm_A_pca = np.linalg.norm(A - np.dot(F, eigvecs_pca[:, :3].T) + mean_A)
print("|A - π(A)| is:", norm_A_pca)
```
首先,读取数据集A.cav,然后使用`np.linalg.svd()`函数进行SVD分解,并将分解后的三个矩阵分别赋值给变量U、S、VT。根据问题要求,依次输出第四个奇异值、A的秩、特征分解的结果、非零特征值的个数、A_k、A-A_k、PCA降维后的最佳3维子空间F,以及|A-A_k|和|A-π(A)|的值。
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