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## Singular Value Decomposition (SVD) (hard)

#### Example

## Singular Value Decomposition (SVD) via the Jacobi Method

### The Jacobi Method for SVD

#### Steps of the Jacobi SVD Algorithm

### Practical Considerations

Write a Python function that approximates the Singular Value Decomposition on a 2x2 matrix by using the jacobian method and without using numpy svd function, i mean you could but you wouldn't learn anything. return the result in this format.

Example: input: a = [[2, 1], [1, 2]] output: (array([[-0.70710678, -0.70710678], [-0.70710678, 0.70710678]]), array([3., 1.]), array([[-0.70710678, -0.70710678], [-0.70710678, 0.70710678]])) reasoning: U is the first matrix sigma is the second vector and V is the third matrix

Singular Value Decomposition (SVD) is a powerful matrix decomposition technique in linear algebra that expresses a matrix as the product of three other matrices, revealing its intrinsic geometric and algebraic properties. When using the Jacobi method, SVD decomposes a matrix \(A\) into:

\[ A = U\Sigma V^T \]- \(A\) is the original \(m \times n\) matrix.
- \(U\) is an \(m \times m\) orthogonal matrix whose columns are the left singular vectors of \(A\).
- \(\Sigma\) is an \(m \times n\) diagonal matrix containing the singular values of \(A\).
- \(V^T\) is the transpose of an \(n \times n\) orthogonal matrix whose columns are the right singular vectors of \(A\).

The Jacobi method is an iterative algorithm used for diagonalizing a symmetric matrix through a series of rotational transformations. It is particularly suited for computing the SVD by iteratively applying rotations to minimize off-diagonal elements until the matrix is diagonal.

**Initialization**: Start with \(A^TA\) (or \(AA^T\) for \(U\)) and set \(V\) (or \(U\)) as an identity matrix. The goal is to diagonalize \(A^TA\), obtaining \(V\) in the process.**Choosing Rotation Targets**: Identify off-diagonal elements in \(A^TA\) to be minimized or zeroed out through rotations.**Calculating Rotation Angles**: For each target off-diagonal element, calculate the angle \(\theta\) for the Jacobi rotation matrix \(J\) that would zero it. This involves solving for \(\theta\) using \(\text{atan2}\) to accurately handle the quadrant of rotation: \[ \theta = 0.5 \cdot \text{atan2}(2a_{ij}, a_{ii} - a_{jj}) \] where \(a_{ij}\) is the target off-diagonal element, and \(a_{ii}\), \(a_{jj}\) are the diagonal elements of \(A^TA\).**Applying Rotations**: Construct \(J\) using \(\theta\) and apply the rotation to \(A^TA\), effectively reducing the magnitude of the target off-diagonal element. Update \(V\) (or \(U\)) by multiplying it by \(J\).**Iteration and Convergence**: Repeat the process of selecting off-diagonal elements, calculating rotation angles, and applying rotations until \(A^TA\) is sufficiently diagonalized.**Extracting SVD Components**: Once diagonalized, the diagonal entries of \(A^TA\) represent the squared singular values of \(A\). The matrices \(U\) and \(V\) are constructed from the accumulated rotations, containing the left and right singular vectors of \(A\), respectively.

- The Jacobi method is particularly effective for dense matrices where off-diagonal elements are significant.
- Careful implementation is required to ensure numerical stability and efficiency, especially for large matrices.
- The iterative nature of the Jacobi method makes it computationally intensive, but it is highly parallelizable.

import numpy as np def svd_2x2_singular_values(A: np.ndarray) -> tuple: A_T_A = A.T @ A theta = 0.5 * np.arctan2(2 * A_T_A[0, 1], A_T_A[0, 0] - A_T_A[1, 1]) j = np.array([[np.cos(theta), -np.sin(theta)], [np.sin(theta), np.cos(theta)]]) A_prime = j.T @ A_T_A @ j # Calculate singular values from the diagonalized A^TA (approximation for 2x2 case) singular_values = np.sqrt(np.diag(A_prime)) # Process for AA^T, if needed, similar to A^TA can be added here for completeness return j, singular_values, j.T

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