Singular Value Decomposition (SVD)

For an \(m \times n\) matrix \(A\), the SVD is a factorization:

\[ A = U\Sigma V^T \]

Components:

• \(U\) is an \(m \times m\) orthogonal matrix (left singular vectors)²
• \(\Sigma\) is an \(m \times n\) diagonal matrix (singular values \(\sigma_1 \geq \sigma_2 \geq \cdots \geq 0\))
• \(V^T\) is an \(n \times n\) orthogonal matrix (right singular vectors)

Key Properties:

• Singular values are the square roots of eigenvalues of \(A^TA\) (or \(AA^T\))³
• The rank of \(A\) equals the number of non-zero singular values
• SVD works for any matrix (square or rectangular)

This calculator uses Jacobi eigendecomposition⁴ of \(A^TA\) to compute the SVD, with Gram-Schmidt orthogonalization to complete the \(U\) matrix columns, and follows the MATLAB sign convention.

To see the rotation–scaling–rotation geometry of \(A = U\Sigma V^T\) animated on the unit circle, open the 2D SVD Visualization.