singulære

Singuläre is the inflected form of the German adjective singulär, meaning singular or unique. In technical language it is used to describe objects that exhibit a degeneracy, exception, or special case within their domain. The form singuläre specifically appears before feminine singular nouns (for example die singuläre Matrix, die singuläre Lösung) and has other endings for different genders and numbers.

In mathematics and related sciences, singulär is used to denote distinct kinds of degeneracy or non-generic behavior. A matrix is singulär if its determinant is zero, which means it is not invertible and has a nontrivial null space; this is referred to as a singuläre Matrix. A singular point on a curve, surface, or more generally a variety is a point where the derivative or the Jacobian matrix fails to have full rank, making the usual tangent structure ill-defined. In differential equations, a singuläre Lösung is a solution that cannot be obtained by simply solving the equation for constants of integration from the general solution. The term also appears in linear algebra in the context of singular values, with singularwerte (singuläre Werte) describing the spectrum in the singular value decomposition.

Outside pure math, singulär is used more broadly to indicate exceptional or non-generic cases in various scientific contexts. The term is etymologically from Latin singularis, and in German it remains closely tied to notions of degeneracy and uniqueness. See also singularity and singular value for related concepts.