Nonsingular

Nonsingular is a term used in mathematics to indicate absence of singularities or, in many contexts, invertibility. The precise meaning depends on the object under consideration. In linear algebra, a square matrix is called nonsingular if it is invertible. In differential geometry and algebraic geometry, a point or a variety is labeled nonsingular (or smooth) if it has no singularities at that point.

Equivalently, for an n-by-n matrix A, the following are true: det(A) ≠ 0; the columns (and rows) are linearly independent; rank(A) = n; A has a unique inverse A^{-1}; the homogeneous system Ax=0 has only the trivial solution; an eigenvalue of A cannot be 0. These conditions ensure a one-to-one mapping and well-posed linear systems.

In calculus and differential geometry, a function or map is nonsingular at a point if the derivative (Jacobian) is of full rank, allowing the inverse function theorem to guarantee local invertibility and smoothness of the mapping in a neighborhood of that point. In this sense, nonsingularity signals that the object behaves well locally rather than having a cusp, cusp-like behavior, or a self-intersection.

In algebraic geometry, a nonsingular (smooth) variety has no cusps or self-intersections; the tangent spaces have the expected dimension at every point, indicating a well-behaved geometric structure.

In statistics, a nonsingular covariance matrix is positive definite and invertible, enabling standard multivariate techniques. In numerical linear algebra, nonsingularity guarantees a unique solution to linear systems and the existence of a (stable) inverse, subject to the conditioning of the problem.

Examples: The matrix [[1,0],[0,1]] is nonsingular with determinant 1 and inverse itself. The matrix [[1,2],[2,4]] is singular since its determinant is 0 and it does not have full rank.