Singular Value Decomposition (2x2)

Background

The Singular Value Decomposition factors any matrix as $A = U\Sigma V^\top$, where $U$ and $V$ are orthogonal and $\Sigma$ is diagonal with non-negative singular values. It is the most general matrix factorization — behind PCA, low-rank approximation, pseudo-inverses, and recommender systems. For a $2\times 2$ matrix it has a closed form via a single Jacobi rotation that diagonalizes $A^\top A$.

Problem statement

Implement svd_2x2_singular_values(A) that returns the SVD $(U, s, V^\top)$ of a $2\times 2$ matrix A. Diagonalize $A^\top A$ with one Jacobi rotation $R$ at angle $\theta$ (so $V = R$), take the singular values as the square roots of the resulting diagonal, then set $U = A V \Sigma^{-1}$:

$$\theta = \tfrac{1}{2}\arctan\!\frac{2\,(A^\top A)_{01}}{(A^\top A)_{00} - (A^\top A)_{11}}, \qquad A = U\,\Sigma\,V^\top$$

(use $\theta = \pi/4$ when the diagonal entries of $A^\top A$ are equal).

Input

• A — np.ndarray of shape (2, 2) (assumed full rank).

Output

Returns a tuple (U, s, V_T): U is (2, 2) left singular vectors, s is (2,) singular values, and V_T is the (2, 2) transpose of the right singular vectors.

Examples

Example 1

Input:  A = [[2, 1], [1, 2]]
Output: U ~= [[-0.7071, -0.7071], [-0.7071, 0.7071]], s = [3, 1],
        V_T ~= [[-0.7071, -0.7071], [-0.7071, 0.7071]]

Explanation: $A^\top A = \begin{bmatrix}5 & 4\\ 4 & 5\end{bmatrix}$ has equal diagonal entries, so $\theta = \pi/4$; the singular values are $\sqrt{9} = 3$ and $\sqrt{1} = 1$, and $U\Sigma V^\top$ reconstructs $A$.

Constraints

• One Jacobi rotation suffices because $A^\top A$ is a symmetric $2\times 2$ matrix.
• The decomposition has a sign ambiguity (columns of $U$/$V$ may flip together); tests check the reconstruction $U\Sigma V^\top = A$ rather than exact entries.
• Singular values are non-negative; A is assumed full rank (no zero singular value).

Notes

• Singular values are the square roots of the eigenvalues of $A^\top A$, and the right singular vectors are that matrix's eigenvectors.
• This $2\times 2$ routine is the kernel of one-sided Jacobi SVD, which sweeps over all column pairs to decompose larger matrices.