Two-by-Two Matrix Decomposition via Jacobi Rotation

The Singular Value Decomposition (SVD) is one of the most powerful and versatile tools in numerical linear algebra, enabling us to decompose any matrix into a product of three special matrices that reveal its fundamental geometric properties.

For a real 2×2 matrix A, the SVD expresses it as:

$$A = U \cdot \Sigma \cdot V^T$$

Where:

• U is a 2×2 orthogonal matrix whose columns are the left singular vectors
• Σ (Sigma) is a diagonal matrix containing the singular values (σ₁ and σ₂)
• V^T is the transpose of the 2×2 orthogonal matrix V, whose columns are the right singular vectors

The Jacobi Rotation Approach:

One elegant method to compute the SVD of a 2×2 matrix involves a Jacobi rotation—a planar rotation designed to diagonalize symmetric matrices. The algorithm works as follows:

1. Form the symmetric product: Compute B = AᵀA (this is always symmetric and positive semi-definite)
2. Compute the Jacobi rotation angle: For the symmetric matrix B with elements b₁₁, b₁₂, b₂₂, calculate the rotation angle θ that diagonalizes B:
   • If b₁₂ = 0, the matrix is already diagonal (θ = 0)
   • Otherwise: θ = 0.5 × arctan(2 × b₁₂ / (b₁₁ - b₂₂))
3. Construct the rotation matrix V: This is an orthogonal matrix built from cos(θ) and sin(θ)
4. Compute singular values: The eigenvalues of B are the squared singular values of A
5. Derive U: Calculate U = A × V × Σ⁻¹ (where Σ⁻¹ contains the reciprocals of singular values)

This single-rotation approach provides an approximate SVD that is exact for symmetric matrices and offers a good approximation for general 2×2 matrices.

Your Task: Implement a Python function that computes the approximate SVD of a real 2×2 matrix using a single Jacobi rotation step. You must not use any high-level SVD library functions (like numpy.linalg.svd). Instead, build the decomposition from fundamental matrix operations.