Type one row per line. Separate entries with spaces, commas, or semicolons. This tool supports rectangular, square, and singular matrices.
This sample uses a 3 × 2 matrix. The pseudoinverse becomes a 2 × 3 matrix because dimensions reverse.
| Row | Input Matrix A | Example Pseudoinverse A+ |
|---|---|---|
| 1 | [1, 2] | [-1.333333, -0.333333, 0.666667] |
| 2 | [3, 4] | [1.083333, 0.333333, -0.416667] |
| 3 | [5, 6] | — |
The calculator uses the Moore–Penrose pseudoinverse. It first forms a singular-value style decomposition from the symmetric matrix ATA.
Main identity: A = UΣVT
Pseudoinverse: A+ = VΣ+UT
Singular-value rule: Every nonzero singular value σ becomes 1/σ. Values below the tolerance are treated as zero.
Verification metric: The calculator reports ‖AA+A − A‖F, the Frobenius-norm reconstruction error.
- Enter the matrix with one row on each line.
- Separate numbers with spaces, commas, or semicolons.
- Choose decimal places, tolerance, and iteration limit.
- Submit the form to calculate the pseudoinverse.
- Review rank, singular values, condition number, and residual checks.
- Use the CSV or PDF buttons to export the results.
1. What is a matrix pseudoinverse?
A pseudoinverse is a generalized inverse for any matrix, including rectangular or singular ones. It supports least-squares solutions, projections, and stable linear-algebra workflows when a normal inverse does not exist.
2. When does the pseudoinverse equal the regular inverse?
If the matrix is square and full rank, the pseudoinverse matches the ordinary inverse. For singular or rectangular matrices, it still exists and provides the Moore–Penrose solution.
3. Why are singular values important here?
Singular values reveal matrix strength, rank, and numerical stability. Very small singular values indicate near-dependence between columns or rows and can make results sensitive to rounding.
4. What does the tolerance setting do?
Tolerance decides which singular values count as effectively zero. Raising it can suppress unstable directions; lowering it can preserve more detail but may amplify numerical noise.
5. What does the residual norm measure?
It measures how closely the reconstructed matrix AA+A matches the original matrix. Smaller values usually indicate a more accurate numerical result.
6. Can I use this for least-squares problems?
Yes. If you want the least-squares solution of Ax = b, you can compute x = A+b. That is a major use of the pseudoinverse.
7. Does this work for non-square matrices?
Yes. The calculator is designed for square, tall, and wide matrices. The pseudoinverse flips dimensions, so an m × n matrix returns an n × m result.
8. Why might the condition number show infinity?
An infinite condition number usually means the matrix is rank-deficient or nearly singular. In that case, some directions collapse and the matrix cannot behave like a stable full-rank inverse problem.