pseudoinversen

The pseudoinversen, also known as the Moore–Penrose pseudoinverse, of a matrix A is a generalization of the inverse that applies to non-square or singular matrices. For A in R^{m×n}, the pseudoinverse is denoted A^+ and satisfies specific algebraic properties that characterize its optimality and projections.

The defining properties, known as the Penrose equations, are AA^+A = A, A^+AA^+ = A^+, (AA^+)^T = AA^+, and (A^+A)^T = A^+A. These conditions ensure that A^+ acts as a best-possible inverse in several senses, and they guarantee uniqueness. In particular, AA^+ is the projection onto the column space (range) of A, while A^+A is the projection onto the row space (co-range) of A.

Computation of the pseudoinverse is commonly performed via singular value decomposition (SVD). If A = U Σ V^T with Σ = diag(σ_i) and σ_i ≥ 0, then A^+ = V Σ^+ U^T, where Σ^+ contains the reciprocals of the nonzero singular values and zeros for the remaining entries. In special cases, closed-form expressions apply: 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}.

The pseudoinverse provides least-squares solutions to Ax ≈ b. Among all solutions, x^* = A^+ b minimizes the Euclidean norm ||x||_2, yielding the minimum-norm solution to the system. It is widely used in data fitting, regression, signal processing, and control theory. For complex matrices, the conjugate transpose is used, denoted A^H.