How this calculator works
The Moore-Penrose pseudoinverse A+ satisfies four identities: A A+ A = A, A+ A A+ = A+, and both products are symmetric. It agrees with the ordinary inverse when one exists, and it gives the minimum-norm least-squares solution of Ax = b when no exact solution exists. This calculator builds it from the singular value decomposition by inverting each non-zero singular value and leaving the zero ones at zero.
Formula
If A = U Sigma V^T, then A+ = V Sigma+ U^T, where Sigma+ has 1/sigma on its diagonal for non-zero singular values and 0 elsewhere.
Worked example
The diagonal matrix with rows 2, 0 and 0, 4 has pseudoinverse 0.5, 0; 0, 0.25. The rectangular matrix with rows 1, 0; 0, 1; 0, 0 has pseudoinverse 1, 0, 0; 0, 1, 0. A rank-one matrix such as rows 1, 2 and 2, 4 has pseudoinverse 0.04, 0.08; 0.08, 0.16.
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Frequently asked questions
When is the pseudoinverse the same as the inverse?
When the matrix is square and invertible. Then all singular values are non-zero and A+ equals A inverse.
Which solution does it produce?
For an inconsistent system, A+ b is the least-squares solution with the smallest norm. For a consistent system with free variables, it selects the solution of smallest norm.
Can every matrix have a pseudoinverse?
Yes. Every matrix, including a zero matrix and a rectangular one, has a unique Moore-Penrose pseudoinverse.