Last updated October 13, 2026 · Report an error
A matrix that isn't square (or is singular) has no true inverse, but it does have a Moore-Penrose pseudoinverse — the closest possible generalization, essential for least-squares problems. This tool computes it via the same SVD engine used by the SVD calculator.
How it works
From the singular value decomposition A = UΣVᵀ, the pseudoinverse is A⁺ = VΣ⁺Uᵀ, where Σ⁺ is built by taking the reciprocal of every non-zero singular value (and leaving any near-zero singular value as 0, since it can't be safely inverted).
- Click Calculate to see your results.
Examples
A rectangular matrix
For a 3x2 matrix (more rows than columns), the pseudoinverse is 2x3 and satisfies A⁺A ≈ the identity matrix.
Who should use it
- Linear algebra coursework on generalized inverses and least squares.
- Data fitting and regression problems needing a least-squares solution.
Industry applications
- Data science and statistics (least-squares regression)
- Robotics and control theory (inverse kinematics)
Advantages
- Reuses the same verified SVD engine as the site's SVD calculator.
- Works for rectangular, square, and singular matrices alike.
Limitations
- Limited to matrices up to 6x6 for reliable, fast computation.
Common mistakes to avoid
- Expecting A⁺A to always equal the identity matrix — that only happens when A has full column rank; otherwise A⁺A is a projection matrix instead.
Best practices
- If you only need the inverse of a genuinely square, non-singular matrix, use a dedicated matrix inverse tool instead — the pseudoinverse is specifically for the non-square or singular case.
Tips
- If A is square and invertible, running it through this tool should reproduce exactly the same result as a regular matrix inverse — a good way to sanity-check the computation.
Frequently asked questions
It's the standard tool for solving least-squares problems (finding the "best fit" solution to an overdetermined or underdetermined system Ax=b that has no exact solution).
For a square, invertible matrix, the pseudoinverse equals the regular inverse exactly — it only becomes a genuinely different (and more useful) object for non-square or singular matrices.