Decomposing a 2×2 Matrix via Singular Value Factorization

Hard

Singular Value Decomposition (SVD) is one of the most powerful and versatile factorization techniques in linear algebra and numerical computing. It decomposes any matrix into a product of three matrices that reveal the matrix's fundamental geometric structure: how it rotates, scales, and rotates again when applied as a linear transformation. For a 2×2 matrix A, the SVD expresses it as: $$A = U \cdot \Sigma \cdot V$$ Where: U is a 2×2 orthogonal matrix containing the left singular vectors (output basis directions) Σ (Sigma) is a 2×2 diagonal matrix with singular values σ₁ ≥ σ₂ ≥ 0 on its diagonal V is a 2×2 matrix containing the right singular vectors (input basis directions) The singular values represent the scaling factors applied to each principal direction. For this problem, we return s as a 1D array [σ₁, σ₂] rather than the full diagonal matrix. Mathematical Foundation: The SVD can be computed by finding the eigenvalue decomposition of AᵀA and AAᵀ: The columns of V are eigenvectors of AᵀA The columns of U are eigenvectors of AAᵀ The singular values are the square roots of the eigenvalues of AᵀA (or equivalently, AAᵀ) Geometric Interpretation: Any linear transformation represented by matrix A can be understood as: Rotate/Reflect (by V): Align with the principal directions Scale (by Σ): Stretch or compress along each axis by σ₁ and σ₂ Rotate/Reflect (by U): Position into the output space Your Task: Write a Python function that computes the SVD of a 2×2 matrix. The function should: Return the orthogonal matrix U (left singular vectors) Return the singular values s as a list of two non-negative numbers Return the matrix V (right singular vectors) Ensure the reconstruction U @ diag(s) @ V approximates the original matrix A Important Constraints: Do not use numpy.linalg.svd or other built-in SVD functions You may use NumPy for basic matrix operations (transpose, multiplication, square roots) Round all outputs to 4 decimal places

Example 1

Input

a = [[-10.0, 8.0], [10.0, -1.0]]

Output

U = [[-0.8, 0.6], [0.6, 0.8]]
s = [15.6525, 4.4721]
V = [[0.8944, -0.4472], [0.4472, 0.8944]]

Explanation

For the matrix A = [[-10, 8], [10, -1]], we compute: 1. **AᵀA** = [[200, -90], [-90, 65]] — eigenvalues: 245.0 and 20.0 2. **Singular values**: √245.0 ≈ 15.6525, √20.0 ≈ 4.4721 3. **V** is constructed from eigenvectors of AᵀA 4. **U** is computed as U = A × V × Σ⁻¹ (or from eigenvectors of AAᵀ) **Verification:** U @ diag(s) @ V ≈ [[-10, 8], [10, -1]] ✓ The larger singular value (15.6525) indicates the primary stretching direction, while the smaller value (4.4721) represents the secondary direction.

Example 2

Input

a = [[1.0, 0.0], [0.0, 1.0]]

Output

U = [[1.0, 0.0], [0.0, 1.0]]
s = [1.0, 1.0]
V = [[1.0, 0.0], [0.0, 1.0]]

Explanation

The **identity matrix** is a special case where: - All singular values are equal to 1.0 - Both U and V are the identity matrix - The decomposition is trivially A = I × diag([1,1]) × I = I This represents a transformation that preserves all vectors exactly as they are — no rotation, no scaling.

Example 3

Input

a = [[3.0, 4.0], [0.0, 5.0]]

Output

U = [[0.7071, -0.7071], [0.7071, 0.7071]]
s = [6.7082, 2.2361]
V = [[0.3162, 0.9487], [-0.9487, 0.3162]]

Explanation

For this upper triangular matrix: 1. **AᵀA** = [[9, 12], [12, 41]] — compute its eigenvalues and eigenvectors 2. **Singular values**: σ₁ ≈ 6.7082, σ₂ ≈ 2.2361 3. The rotation angle in U is 45° (hence the √2/2 ≈ 0.7071 values) This decomposition reveals that the matrix stretches vectors by about 6.7× in its primary direction.

Constraints

Input matrix is always exactly 2×2
-10⁶ ≤ a[i][j] ≤ 10⁶ for all matrix elements
The matrix may contain integers or floating-point numbers
Singular values should be non-negative and returned in descending order
Output values should be rounded to 4 decimal places
Do not use numpy.linalg.svd or scipy.linalg.svd
The reconstruction U @ diag(s) @ V should closely approximate the original matrix