🔄 Pseudoinverse Calculator
Calculate Moore-Penrose pseudoinverse A⁺
How to Use This Calculator
Enter Matrix
Input any matrix (can be rectangular or singular).
Calculate
Get Moore-Penrose pseudoinverse A⁺.
Formula
A⁺ = (AᵀA)⁻¹Aᵀ (for full column rank)
Definition:
Moore-Penrose pseudoinverse A⁺ satisfies: AA⁺A = A, A⁺AA⁺ = A⁺, (AA⁺)ᵀ = AA⁺, (A⁺A)ᵀ = A⁺A.
Properties:
- If A is invertible: A⁺ = A⁻¹
- If A has full column rank: A⁺ = (AᵀA)⁻¹Aᵀ
- If A has full row rank: A⁺ = Aᵀ(AAᵀ)⁻¹
- General case: A⁺ = VΣ⁺Uᵀ (via SVD)
About Pseudoinverse Calculator
The pseudoinverse calculator computes Moore-Penrose pseudoinverse A⁺, a generalization of matrix inverse for rectangular and singular matrices. It minimizes least-squares errors and is unique for any matrix.
When to Use This Calculator
- Least Squares: Solve overdetermined systems
- Singular Systems: Find solutions when A is not invertible
- Linear Regression: Normal equations solution
- Moore-Penrose: General matrix equations
Why Use Our Calculator?
- ✅ Moore-Penrose: Standard pseudoinverse
- ✅ Works for All: Rectangular, singular, or invertible matrices
- ✅ Educational: Helps understand pseudoinverse
- ✅ Free: No registration required
Key Concepts
- Moore-Penrose: Unique pseudoinverse satisfying 4 conditions
- Least Squares: A⁺b is minimizer of ||Ax - b||²
- Rank: Rank(A⁺) = Rank(A)
- SVD Method: Most general computation method
Frequently Asked Questions
What is pseudoinverse?
Generalization of inverse for all matrices. For invertible matrices, A⁺ = A⁻¹. For others, it provides least-squares optimal solution.
Why use pseudoinverse instead of inverse?
Pseudoinverse exists for every matrix, even rectangular or singular. Inverse only exists for square nonsingular matrices.
How is pseudoinverse related to SVD?
If A = UΣVᵀ (SVD), then A⁺ = VΣ⁺Uᵀ where Σ⁺ has inverse of nonzero singular values.
Is pseudoinverse unique?
Yes, Moore-Penrose pseudoinverse is unique for any matrix, defined by the 4 Moore-Penrose conditions.