Enter your matrix values into the grid and choose the matrix dimensions — the Pseudoinverse Calculator computes the Moore-Penrose pseudoinverse (A⁺) of any matrix up to 4×4. Set your rows and columns, fill in the matrix entries, and get back the full pseudoinverse matrix along with intermediate values like the transpose and rank. Also try the Diagonalize Matrix Calculator.
Pseudoinverse A⁺
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Rank of A
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Formula Used
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Frobenius Norm of A⁺
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The Moore-Penrose pseudoinverse (A⁺) is a generalization of the matrix inverse that works for any matrix, including non-square and singular matrices. It satisfies four conditions known as the Moore-Penrose conditions and is widely used in solving least-squares problems and linear systems that may have no unique solution. See also our calculate Matrix Norm Frobenius Norm ‖A‖_F.
A regular inverse exists only for square, non-singular matrices (full rank). The pseudoinverse always exists for any matrix — square or rectangular, full rank or not. For a square invertible matrix, A⁺ equals A⁻¹ exactly.
For a wide matrix (more columns than rows, i.e. m < n), the right pseudoinverse A⁺ = Aᵀ(AAᵀ)⁻¹ is used. For a tall matrix (more rows than columns, i.e. m > n), the left pseudoinverse A⁺ = (AᵀA)⁻¹Aᵀ is used. For square full-rank matrices, this reduces to A⁻¹.
Singular Value Decomposition (SVD) decomposes A into U Σ Vᵀ. The pseudoinverse is then A⁺ = V Σ⁺ Uᵀ, where Σ⁺ is formed by taking the reciprocal of each non-zero diagonal entry in Σ and transposing. SVD-based computation is the most numerically stable method for the pseudoinverse. You might also find our calculate Characteristic Polynomial p(λ), Trace of Matrix (sum of diagonal) & Determinant of Matrix — Characteristic Polynomial useful.
The rank is the number of linearly independent rows or columns in the matrix. It determines how many non-zero singular values exist and affects which pseudoinverse formula applies. A rank-deficient matrix has fewer independent directions, so the pseudoinverse projects onto only those directions.
Yes. For a system Ax = b, the minimum-norm least-squares solution is x = A⁺b. If the system is consistent, this gives the exact solution with the smallest norm. If it is inconsistent (over-determined), it gives the best approximate solution that minimizes ‖Ax − b‖².
This calculator supports matrices from 1×1 up to 4×4. You can enter any combination of rows (1–4) and columns (1–4). Only the entries corresponding to your chosen dimensions are used; the rest are ignored.
The Frobenius norm of A⁺ is the square root of the sum of squares of all entries in A⁺. It gives an overall magnitude measure of the pseudoinverse matrix and can be useful for assessing numerical conditioning.