pseudoinverse

The Moore–Penrose pseudoinverse of a matrix A ∈ R^{m×n} is the unique matrix A^+ ∈ R^{n×m} that satisfies the four Penrose equations: A A^+ A = A, A^+ A A^+ = A^+, (A A^+)^T = A A^+, and (A^+ A)^T = A^+ A. It generalizes the inverse to non-square or singular matrices and serves as a means to obtain least-squares solutions to Ax ≈ b.

For a given vector b, the solution x̂ = A^+ b is the least-squares solution with minimum Euclidean norm among all solutions to Ax ≈ b. If A is square and nonsingular, A^+ equals the ordinary inverse A^{-1}.

Special cases arise when A has full rank. If A has full column rank (m ≥ n, rank n), then A^+ = (A^T A)^{-1} A^T. If A has full row rank (n ≥ m, rank m), then A^+ = A^T (A A^T)^{-1}.

Computation is commonly performed via the singular value decomposition (SVD). Writing A = U Σ V^T, where Σ = diag(σ_1, ..., σ_r, 0, ..., 0) with r = rank(A), the pseudoinverse is A^+ = V Σ^+ U^T, where Σ^+ replaces each nonzero σ_i by 1/σ_i and transposes the diagonal shape. This formulation handles rank-deficient and ill-conditioned cases.

Applications span solving inconsistent linear systems, data fitting and regression, signal processing, and control theory. The Moore–Penrose pseudoinverse is unique among generalized inverses and specializes to the ordinary inverse when A is square and nonsingular. Regularization methods (e.g., Tikhonov) can be interpreted as modifying the pseudoinverse to stabilize solutions.