고윳값과 SVD 개요

고유값과 고유벡터란 무엇인가요?

import numpy as np

A = np.array([[3.0, 1.0],
              [0.0, 2.0]])

eigenvalues, eigenvectors = np.linalg.eig(A)
print('Eigenvalues:', eigenvalues)
print('Eigenvectors (columns):')
print(eigenvectors)

np.linalg.eig()로 고유값 계산하기

import numpy as np

# Symmetric matrix -> use eigh for efficiency and real eigenvalues
A = np.array([[4.0, 2.0],
              [2.0, 3.0]])

vals, vecs = np.linalg.eigh(A)
print('Eigenvalues (real):', vals)
print('Eigenvectors (orthonormal columns):')
print(vecs)

# Verify: A @ v = lambda * v for each eigenvector
for i in range(len(vals)):
    lhs = A @ vecs[:, i]
    rhs = vals[i] * vecs[:, i]
    print(f'v{i} check:', np.allclose(lhs, rhs))

고유값과 행렬의 특성

import numpy as np

A = np.array([[4.0, 2.0],
              [2.0, 3.0]])

vals, _ = np.linalg.eigh(A)
print('Eigenvalues:', vals)
print('Product (should = det):', np.prod(vals))
print('np.linalg.det:', np.linalg.det(A))
print('Sum (should = trace):', np.sum(vals))
print('np.trace:', np.trace(A))
print('Positive definite:', np.all(vals > 0))

특이값 분해(SVD) 개요

import numpy as np

A = np.array([[1.0, 2.0, 3.0],
              [4.0, 5.0, 6.0]])

U, s, Vh = np.linalg.svd(A, full_matrices=False)
print('U shape:', U.shape)
print('s (singular values):', s)
print('Vh shape:', Vh.shape)

# Reconstruct A
A_reconstructed = U @ np.diag(s) @ Vh
print('Reconstruction correct:', np.allclose(A, A_reconstructed))

특이값과 행렬 계수

import numpy as np

# Full-rank matrix
A = np.random.rand(5, 3)
U, s, Vh = np.linalg.svd(A, full_matrices=False)
print('Singular values:', np.round(s, 4))
print('Rank (non-zero sv):', np.linalg.matrix_rank(A))
print('Condition number:', s[0] / s[-1])

# Rank-deficient matrix
B = np.array([[1.0, 2.0], [2.0, 4.0], [3.0, 6.0]])
_, sb, _ = np.linalg.svd(B, full_matrices=False)
print('Rank-deficient sv:', np.round(sb, 8))

SVD와 PCA의 연관성

import numpy as np

np.random.seed(0)
X = np.random.randn(100, 4)

# Center the data
X_centered = X - X.mean(axis=0)

# SVD-based PCA
U, s, Vh = np.linalg.svd(X_centered, full_matrices=False)

# Variance explained by each component
var_explained = (s ** 2) / (X.shape[0] - 1)
total_var = var_explained.sum()
print('Variance explained ratio:', np.round(var_explained / total_var, 3))

# Project onto top 2 principal components
X_pca = X_centered @ Vh[:2].T
print('Reduced shape:', X_pca.shape)

SVD를 사용한 저계수 근사

import numpy as np

np.random.seed(1)
A = np.random.rand(20, 15)

U, s, Vh = np.linalg.svd(A, full_matrices=False)

# Rank-3 approximation
k = 3
A_approx = U[:, :k] @ np.diag(s[:k]) @ Vh[:k, :]

error = np.linalg.norm(A - A_approx, 'fro')
total = np.linalg.norm(A, 'fro')
print(f'Approximation error: {error/total:.3f} (fraction of total)')
print(f'Top-3 singular values capture {(s[:3]**2).sum()/(s**2).sum():.1%} of variance')

고유분해와 SVD: 어느 것을 사용해야 할까요?

import numpy as np

# Non-symmetric matrix: eigenvalues may be complex
A = np.array([[0.0, -1.0],
              [1.0,  0.0]])
vals, _ = np.linalg.eig(A)
print('Eigenvalues (complex for rotation):', vals)

# SVD always gives real singular values
_, s, _ = np.linalg.svd(A)
print('Singular values (always real):', s)

대칭 행렬의 스펙트럼 정리

import numpy as np

A = np.array([[5.0, 2.0, 1.0],
              [2.0, 3.0, 0.0],
              [1.0, 0.0, 4.0]])

vals, Q = np.linalg.eigh(A)
print('Eigenvalues:', np.round(vals, 4))

# Reconstruct A = Q @ diag(vals) @ Q.T
A_reconstructed = Q @ np.diag(vals) @ Q.T
print('Reconstruction correct:', np.allclose(A, A_reconstructed))
print('Q is orthogonal:', np.allclose(Q.T @ Q, np.eye(3)))

실습: 지배적 고유값을 구하는 거듭제곱 반복법

import numpy as np

A = np.array([[4.0, 1.0, 2.0],
              [1.0, 3.0, 0.0],
              [2.0, 0.0, 2.0]])

v = np.random.rand(3)
for _ in range(50):
    v = A @ v
    v = v / np.linalg.norm(v)

eigenvalue_approx = v @ A @ v
print('Dominant eigenvector:', np.round(v, 4))
print('Approx eigenvalue:', round(eigenvalue_approx, 4))

# Verify with eigh
vals, vecs = np.linalg.eigh(A)
print('True max eigenvalue:', round(vals[-1], 4))

실제 파이프라인에서 np.linalg.svd 사용하기

import numpy as np

np.random.seed(7)
X_train = np.random.randn(200, 50)
X_test = np.random.randn(40, 50)

# Fit on training data
X_mean = X_train.mean(axis=0)
X_centered = X_train - X_mean
U, s, Vh = np.linalg.svd(X_centered, full_matrices=False)

# Choose k to capture 90% variance
cumvar = np.cumsum(s**2) / (s**2).sum()
k = np.searchsorted(cumvar, 0.9) + 1
print(f'Components to capture 90% variance: {k}')

# Transform test data using the same Vh
X_test_reduced = (X_test - X_mean) @ Vh[:k].T
print('Reduced test shape:', X_test_reduced.shape)

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