noninvertible

Noninvertible describes objects that do not possess a unique inverse. In mathematics, the term is most commonly applied to square matrices and to functions. A square matrix is noninvertible (also called singular) when there is no matrix B such that AB = BA = I. This occurs exactly when its determinant is zero or, equivalently, when its columns are linearly dependent and its rank is less than the dimension. For a noninvertible matrix A, the linear transformation x -> Ax is not bijective: its kernel is nontrivial, meaning there exist nonzero vectors x with Ax = 0, and some values in the codomain have no unique preimage. Consequently, systems of linear equations Ax = b may have no solution or infinitely many solutions, rather than a unique one.

As a function: a function f: X -> Y is invertible if and only if it is bijective. If f is not injective, or not surjective, it is noninvertible: there is no single inverse function f^{-1}: Y -> X. For example, x^2 as a function R -> R is noninvertible because both 1 and -1 map to 1. Restricting its domain to nonnegative numbers makes it invertible.

In numerical contexts, a noninvertible matrix does not have a true inverse, but a Moore-Penrose pseudoinverse provides a best-fit generalized inverse suitable for solving least-squares problems. The concept is closely related to eigenvalues: a matrix is noninvertible precisely when zero is an eigenvalue.