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SVD of a 2x2 Matrix

#28 · Linear Algebra · Hard

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Problem

Compute the Singular Value Decomposition of a 2x2 matrix manually, returning U, Sigma, and V^T such that A = U Sigma V^T.

Solution

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import numpy as np

def svd_2x2(A: list[list[float]]) -> tuple:
    A = np.array(A, dtype=float)

    # Compute A^T A
    ATA = A.T @ A

    # Eigendecomposition of A^T A
    eigenvalues, V = np.linalg.eigh(ATA)

    # Sort eigenvalues descending
    idx = np.argsort(eigenvalues)[::-1]
    eigenvalues = eigenvalues[idx]
    V = V[:, idx]

    # Singular values
    singular_values = np.sqrt(np.maximum(eigenvalues, 0))
    Sigma = np.diag(singular_values)

    # Compute U
    U = np.zeros((2, 2), dtype=float)
    for i in range(2):
        if singular_values[i] > 1e-10:
            U[:, i] = (A @ V[:, i]) / singular_values[i]
        else:
            # For zero singular values, pick an orthogonal vector
            if i > 0:
                U[:, i] = np.array([-U[1, 0], U[0, 0]])

    # Ensure right-handed coordinate systems
    if np.linalg.det(U) < 0:
        U[:, -1] *= -1
    if np.linalg.det(V) < 0:
        V[:, -1] *= -1

    return (
        np.round(U, 4).tolist(),
        np.round(Sigma, 4).tolist(),
        np.round(V.T, 4).tolist()
    )

Explanation

  1. Compute A^T * A, whose eigenvalues are the squared singular values and eigenvectors form V.
  2. Take the square root of eigenvalues to get singular values; build the diagonal matrix Sigma.
  3. Compute U columns via u_i = A * v_i / sigma_i.
  4. Handle edge cases for zero singular values and ensure proper rotation matrices.

Complexity

  • Time: O(1) for 2x2 matrices
  • Space: O(1)
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