inversez

Inversez is a term used in numerical linear algebra and inverse problem theory to denote a family of inverse-like operators that depend on a parameter z. It is closely related to Tikhonov regularization and generalized inverses, providing a bridge between exact inverses and the Moore–Penrose pseudoinverse.

For a matrix A in R^{m×n} and a real z > 0, the inversez operator is defined as inv_z(A) = (A^T A + z I)^{-1} A^T. This operator yields a stable solution to the least-squares problem min_x ||Ax - b||^2 and acts as a regularized pseudoinverse. As z → 0+, inv_z(A) tends to the Moore-Penrose pseudoinverse A^+ under standard conditions.

Properties and computation: A^T A + z I is symmetric positive definite for z > 0, ensuring existence of inv_z(A). The method is equivalent to ridge regression and introduces bias controlled by z. It can be computed via Cholesky decomposition of A^T A + z I or via singular value decomposition, with z improving conditioning for ill-conditioned A.

Applications and notes: Inversez is used in solving inverse problems in imaging, deconvolution, geophysics, and machine learning, particularly where stable inversion is needed in the presence of noise. The term "inversez" is not universal; it reflects one naming of this regularized inverse concept, with alternatives including Tikhonov regularization or ridge-regularized pseudoinverse.

---