Nietsingulier

Nietsingulier is a Dutch mathematical term used to describe objects that are not singular. In general, it denotes non-singularity, meaning that the object behaves regularly in a sense that allows certain standard properties and theorems to apply.

In linear algebra, a square matrix is nietsingulier if its determinant is nonzero. Such a matrix is invertible, has full rank, and the linear system Ax = b has a unique solution for every right-hand side b. Conversely, a singular matrix has determinant zero and does not have a unique inverse.

In algebraic and differential geometry, a point on a curve, surface, or more generally a variety is nietsingulier if it is a non-singular (smooth) point. The practical criterion is that the Jacobian (the matrix of partial derivatives) has full rank at that point; equivalently, the tangent space has the expected dimension. Singular points occur where the Jacobian loses rank, leading to features such as cusps or self-intersections.

Examples help illustrate the concept. The 2x2 identity matrix is nietsingulier, with determinant 1. The matrix [[1, 0], [0, 0]] is singular, with determinant 0. A non-singular point on the circle x^2 + y^2 = 1 exists at any point on the circle, since the gradient (2x, 2y) is never (0, 0) on that curve.

Nietsingulier underpins many results in analysis and geometry, ensuring invertibility, smooth structure, and well-defined local behavior. See also singularity, smoothness, and the Jacobian criterion.