Singular Value Decomposition (SVD)

Compute the Singular Value Decomposition1 of a matrix \(A = U\Sigma V^T\).
Matrix entries can be real numbers or fractions.

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.

Set matrix dimensions

Enter dimensions and click Generate Matrix to create the input grid.

SVD Type:
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Compute
Computes the full singular value decomposition A = UΣVᵀ.
Resets all dimensions, matrix entries, and results.