nichtunitären

Nichtunitären is a German term that translates to "non-unitary" in English. In mathematics, particularly in the study of linear algebra and functional analysis, it refers to operators or matrices that do not possess a unitary inverse. A unitary operator or matrix is one whose inverse is also its conjugate transpose, meaning U*U = UU* = I, where U* is the conjugate transpose of U and I is the identity operator or matrix. This property ensures that unitary operators preserve inner products and norms, which is crucial in many physical theories, such as quantum mechanics.

Operators or matrices that are non-unitary, therefore, do not have this property. They may still be invertible, but their inverse will not be their conjugate transpose. This means that applying a non-unitary operator can change the lengths of vectors and the angles between them. Non-unitary operators can arise in various contexts, including certain types of transformations, physical systems that are not strictly conserving probability, or in numerical approximations where exact unitarity might be lost. The study of non-unitary operators is important for understanding a broader range of mathematical and physical phenomena where the constraints of unitarity are relaxed.