Pseudoinverse Calculator
Calculate Moore-Penrose pseudoinverse for any matrix with step-by-step solutions
Matrix Input
Select matrix dimensions. Currently optimized for 2×2, 2×3, and 3×2 matrices.
Enter matrix elements. Use decimal numbers for fractional values.
Pseudoinverse Results
Original Matrix A:
Pseudoinverse A⁺:
Method Used:
Regular inverse (matrix is invertible)
Note: Since the matrix is invertible, the pseudoinverse equals the regular inverse.
Properties Verified:
- • A⁺ has dimensions 2×2
- • The pseudoinverse is a generalization of the matrix inverse
- • Satisfies the Moore-Penrose conditions
- • For invertible matrices: A⁺ = A⁻¹
Example: 2×3 Matrix
Try this example:
Matrix A = [[1, 3, 2], [4, 3, 3]]
This matrix has linearly independent columns, so we can use A⁺ = (AᵀA)⁻¹Aᵀ
Calculation Methods
Left Pseudoinverse
A⁺ = (AᵀA)⁻¹Aᵀ
For linearly independent columns
Right Pseudoinverse
A⁺ = Aᵀ(AAᵀ)⁻¹
For linearly independent rows
SVD Method
A⁺ = VS⁺Uᵀ
General case using SVD
Properties
Generalizes the concept of matrix inverse
Exists for any matrix (including non-square)
A⁺ has transposed dimensions of A
If A is invertible, then A⁺ = A⁻¹
Used for solving overdetermined systems
Understanding the Moore-Penrose Pseudoinverse
What is the Pseudoinverse?
The Moore-Penrose pseudoinverse A⁺ is a generalization of the matrix inverse for matrices that may not be invertible. It exists for any matrix and provides the "best" solution to linear systems that may have no solution or infinitely many solutions.
Key Properties
- •Universal existence: Every matrix has a pseudoinverse
- •Dimension property: (m×n) matrix has (n×m) pseudoinverse
- •Reduces to inverse: A⁺ = A⁻¹ when A is invertible
- •Least squares: Minimizes ||Ax - b||² in overdetermined systems
Calculation Methods
Case 1: Linearly independent columns
A⁺ = (AᵀA)⁻¹Aᵀ
Case 2: Linearly independent rows
A⁺ = Aᵀ(AAᵀ)⁻¹
General case (SVD)
A = USVᵀ → A⁺ = VS⁺Uᵀ
Applications
- Linear regression: Finding best-fit lines and curves
- Data fitting: Approximating solutions to inconsistent systems
- Image processing: Reconstruction and denoising
- Machine learning: Principal component analysis
- Control theory: System identification and optimization