Pseudoinverse Calculator
Calculate the Moore-Penrose pseudoinverse of a matrix using SVD decomposition.
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Moore-Penrose Pseudoinverse
The pseudoinverse A⁺ of a matrix A satisfies the following properties:
- AA⁺A = A
- A⁺AA⁺ = A⁺
- (AA⁺)* = AA⁺
- (A⁺A)* = A⁺A
Calculation Methods:
- Singular Value Decomposition (SVD)
- QR Decomposition
- Normal Equations
Applications:
- Least squares problems
- Linear regression
- Data compression
- Image processing
- Control systems
Properties:
- Unique for any matrix
- Reduces to regular inverse for square, non-singular matrices
- Preserves matrix dimensions
- Useful for rank-deficient matrices
Geometric Interpretation:
- Minimizes least squares error
- Projects onto column space
- Handles over/under-determined systems
- Provides optimal solution