Pseudoinverse Calculator

Your details

Number of rows in the input matrix A (m in the m x n notation).
Number of columns in the input matrix A (n in the m x n notation).
Enter values row by row. Separate columns with commas and rows with semicolons. For a 3x2 matrix: 1,2;3,4;5,6
Pseudoinverse A+
-1.3333 -0.3333 0.6667 1.0833 0.3333 -0.4167

Moore-Penrose pseudoinverse of the input matrix A

Method usedFull column rank: A+ = (A^T A)^-1 A^T
Penrose condition A A+ A = ASatisfied (A A+ A = A, error < 1e-6)
Pseudoinverse dimensions2 x 3
Frobenius norm of A9.5394
Frobenius norm of A9.5394

Pseudoinverse computed via Full column rank.

  • The matrix is rank-deficient (neither A^T A nor A A^T is invertible), so the full SVD-based pseudoinverse was used.
  • Singular values below the numerical threshold are treated as zero: the pseudoinverse still exists and satisfies all four Penrose conditions.
  • The pseudoinverse A+ has dimensions 2 x 3 (columns and rows of A are swapped). It maps from the output space back to the input space of A.

Next stepTo solve a system Ax = b, multiply A+ by b: x = A+ b. For overdetermined systems this minimises the residual Ax - b; for underdetermined systems it finds the shortest solution vector.

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