90. Pseudoinverse
The Moore–Penrose pseudoinverse generalizes the matrix inverse to every matrix — square, rectangular, full-rank, or rank-deficient.
For invertible square , . For other matrices, provides the “best possible inverse” in a precise least-squares sense.
90.1. Definition via SVD
If is the SVD of , then:
where is the transpose of with each non-zero singular value replaced by (zero singular values stay zero).
90.2. Four defining properties
is uniquely characterized by the Moore–Penrose conditions:
- — symmetric
- — symmetric
These force a unique for every matrix.
90.3. Special cases
Full column rank (, ):
Full row rank (, ):
Square invertible: .
90.4. Least-squares solution
For the overdetermined system (more equations than unknowns), the minimum-norm least-squares solution is:
Properties:
- If a solution to exists, is one (the one with smallest norm)
- If no exact solution exists, minimizes
This is the foundation of linear regression and least-squares fitting.
Example
No exact solution (rows 2 and 3 are inconsistent with non-zero entries).
SVD: , . So (transpose of with left zero).
This minimizes and has minimum norm among minimizers.
90.5. Why it generalizes the inverse
- Replaces “exact solution” with “best least-squares approximation”
- Handles rank-deficient systems (where standard inverse is undefined)
- Handles rectangular systems (where standard inverse is undefined)
- Reduces to ordinary inverse exactly when ordinary inverse exists
90.6. See also
- SVD — the construction
- Matrix Inverse — special case
- QR — used to compute the pseudoinverse in the full-rank case
- Linear Regression — pseudoinverse gives OLS coefficients