Moore-Penrose Pseudoinverse Calculator

Compute the generalized inverse1 (Moore-Penrose pseudoinverse) \(A^{\dagger}\) using SVD2.
Works for any matrix: square, rectangular, singular, or non-singular.

The Moore-Penrose pseudoinverse \(A^{\dagger}\) generalizes the matrix inverse:

  • If \(A\) is square and invertible: \(A^{\dagger} = A^{-1}\)
  • If \(A\) has full column rank: \(A^{\dagger} = (A^TA)^{-1}A^T\) (left inverse)
  • If \(A\) has full row rank: \(A^{\dagger} = A^T(AA^T)^{-1}\) (right inverse)
  • For any matrix: computed via SVD as \(A^{\dagger} = V\Sigma^{\dagger}U^T\)3

Applications:

  • Solving least squares problems: \(\min \|Ax - b\|\)
  • Finding minimum norm solutions to \(Ax = b\)
  • Computing best-fit approximations
Set matrix dimensions

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

Display values as:
1 — Compute
Computes the inverse (if square and nonsingular) or the Moore–Penrose pseudoinverse.
Resets all dimensions, matrix entries, and results.
Cite this tool
DOI: https://doi.org/10.5281/zenodo.20981225
Kapita, S. (2026). Moore-Penrose Pseudoinverse Calculator. Math Tools. https://doi.org/10.5281/zenodo.20981225