Pseudoinverses

A pseudoinverse is a generalization of the matrix inverse that applies to non-square or singular matrices. The most common form is the Moore–Penrose pseudoinverse, denoted A+, for a real or complex matrix A. It serves as a best-approximation inverse in several senses and underpins many linear-algebra problems where a true inverse does not exist.

The Moore–Penrose pseudoinverse is the unique matrix X that satisfies the four Penrose equations: A X A = A, X A X = X, (A X)^T = A X, and (X A)^T = X A. It exists for every matrix A and reduces to the ordinary inverse when A is square and invertible. The pseudoinverse thus unifies inverses across square, rectangular, full-rank, and rank-deficient cases.

Computation is most efficiently done via the singular value decomposition: A = U Σ V^T, with Σ containing the nonnegative singular values σ_i. Then A+ = V Σ+ U^T, where Σ+ has reciprocal entries 1/σ_i for nonzero σ_i and zeros for zero σ_i. Special cases include: if A has full column rank, A+ = (A^T A)^{-1} A^T; if A has full row rank, A+ = A^T (A A^T)^{-1}.

Applications include solving linear systems in a least-squares sense and obtaining the minimum-norm solution x = A+ b when solving Ax = b. Pseudoinverses are widely used in data fitting, statistics, signal processing, and control theory. Regularization methods, such as Tikhonov regularization, modify the pseudoinverse to improve stability for ill-posed problems. Related concepts include generalized inverses, of which the Moore–Penrose form is a specific, well-behaved instance.