MoorePenrose

MoorePenrose refers to the Moore-Penrose pseudoinverse of a matrix, denoted A^+. It generalizes the matrix inverse to non-square or singular matrices and is the unique matrix that satisfies the four Penrose equations: AA^+A = A, A^+AA^+ = A^+, (AA^+)^T = AA^+, and (A^+A)^T = A^+A. The term is named after E. H. Moore and Roger Penrose and is commonly called the pseudoinverse or generalized inverse.

Computation of the Moore-Penrose inverse is most commonly done via the singular value decomposition. If A = U Σ V^T, where Σ contains the nonnegative singular values σ_i, then A^+ = V Σ^+ U^T, with Σ^+ formed by taking reciprocals of the nonzero σ_i and transposing the diagonal. When A has full column rank, A^+ = (A^T A)^{-1} A^T; when A has full row rank, A^+ = A^T (A A^T)^{-1}.

Properties and applications: The Moore-Penrose inverse provides least-squares solutions to linear systems, with x = A^+ b minimizing ||b − Ax||_2 and having minimal norm among all minimizers. It also yields orthogonal projections, with AA^+ projecting onto the column space of A and A^+A projecting onto the row space of A. It is widely used in statistics, signal processing, control theory, and machine learning for solving linear systems, regression, and data analysis tasks.

The Moore-Penrose inverse exists for every real or complex matrix and is unique. It extends to operators on inner product spaces and has several generalizations, but remains the standard tool for analyzing and solving broad classes of linear problems.