How to Use Pseudoinverse Calculator
The Pseudoinverse Calculator computes A⁺, the Moore-Penrose pseudoinverse, for any real matrix. Enter the matrix dimensions and values, and the Pseudoinverse Calculator automatically selects the appropriate formula, shows the computation method, and outputs the pseudoinverse matrix together with step-by-step derivation.
- Set dimensions – Enter rows and columns (1–5 each).
- Fill in matrix A – Type each element. Negative and decimal values are accepted.
- Read A⁺ – The Pseudoinverse Calculator displays the pseudoinverse matrix and the method used.
- Check the steps – Review the intermediate matrices (AᵀA, (AᵀA)⁻¹, etc.) shown in the steps panel.
- Apply to a system – Use x = A⁺b to find the minimum-norm least-squares solution to Ax = b.
Formula & Theory - Pseudoinverse Calculator
The Pseudoinverse Calculator implements the three standard closed-form formulas for the Moore-Penrose pseudoinverse:
Case 1: A is invertible (square, full rank)
A⁺ = A⁻¹
Case 2: A has full column rank (rank = n ≤ m)
A⁺ = (AᵀA)⁻¹ Aᵀ
Case 3: A has full row rank (rank = m ≤ n)
A⁺ = Aᵀ (AAᵀ)⁻¹
Case 4: A is rank-deficient (neither full row nor column rank)
A⁺ ≈ (AᵀA + εI)⁻¹ Aᵀ (Tikhonov regularization)| Symbol | Meaning |
|---|---|
| A⁺ | Moore-Penrose pseudoinverse |
| AᵀA | Gram matrix (cols × cols) |
| AAᵀ | Dual Gram matrix (rows × rows) |
| ε | Small regularization constant (10⁻⁹) |
| r | rank(A) |
The pseudoinverse satisfies the four Penrose conditions: AA⁺A = A, A⁺AA⁺ = A⁺, (AA⁺)ᵀ = AA⁺, and (A⁺A)ᵀ = A⁺A. The minimum-norm least-squares solution to Ax = b is x = A⁺b, which minimizes ||Ax − b||₂ and among all such solutions also minimizes ||x||₂.
Assumptions and Limits
The Pseudoinverse Calculator supports matrices up to 5×5. Rank is computed via Gaussian elimination with a zero-threshold of 10⁻¹⁰. For rank-deficient cases, a small regularization (ε = 10⁻⁹) is added to AᵀA to make it invertible; this is an approximation and may differ slightly from the exact pseudoinverse computed via SVD.
Use Cases for Pseudoinverse Calculator
The Pseudoinverse Calculator is essential in many areas of applied mathematics and data science:
- Least-squares regression – Solve overdetermined systems Ax = b (more equations than unknowns) using x = A⁺b.
- Underdetermined systems – Find the minimum-norm solution when there are more unknowns than equations.
- Machine learning – Compute weights for linear models without inverting an ill-conditioned matrix.
- Control theory – Design minimum-energy control inputs using the pseudoinverse of the controllability matrix.
- Image reconstruction – Invert degradation operators in deconvolution problems.
After computing with the Pseudoinverse Calculator, you can substitute A⁺ directly into x = A⁺b to solve any compatible linear system in the least-squares sense.