MoorePenroseinverse

The Moore-Penrose inverse, denoted A^+, is a generalized matrix inverse defined for any m-by-n matrix A over the real or complex numbers. It is the unique matrix that provides a best possible solution to linear systems that are inconsistent or ill-conditioned and plays a central role in least-squares problems.

It is characterized by four Penrose equations: A A^+ A = A, A^+ A A^+ = A^+, (A A^+)^T = A A^+, and (A^+ A)^T = A^+ A. When A is invertible, A^+ equals the ordinary inverse; for complex matrices conjugate transpose is used in place of the transpose.

Computation: the most common method is via the singular value decomposition A = U Σ V^*, with Σ = diag(σ_i). Then A^+ = V Σ^+ U^*, where Σ^+ contains reciprocals of the nonzero σ_i and zeros for zero singular values. In special cases: 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}. In numerical work small singular values may be truncated to improve stability.

Applications and notes: A^+ provides the least-squares solution x to Ax ≈ b as x = A^+ b; among all least-squares solutions it has minimal Euclidean norm. It is widely used in statistics, signal processing, control theory, machine learning, and data fitting; it extends to linear operators on Hilbert spaces.