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.
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.
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.
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.