Unitären

Unitären is the inflected form of the German adjective unitär, meaning unitary. It is used in mathematics and physics to describe objects or structures that preserve certain inner products or norms, and it appears in declined forms such as unitären, unitäre, or unitäres depending on case, number, and gender.

In mathematics, unitär denotes relations to the unitary concept. Unitäre Gruppen (unitary groups) consist of unitary matrices under multiplication. Unitäre Matrizen (unitary matrices) are square matrices U over the complex numbers that satisfy U* U = I, where U* is the conjugate transpose of U. This condition implies preservation of the inner product and norm in complex vector spaces. In operator theory and functional analysis, unitär operators are central, and unitäre Darstellungen (unitary representations) describe group actions on Hilbert spaces that preserve the inner product.

In physics, unitär is used to express the principle that evolution preserves probability: time evolution in quantum mechanics is described by unitary operators. In quantum computing, quantum gates are modeled as unitary operations to ensure probability conservation.

Outside mathematics and physics, the form appears in broader German usage related to the notion of unity or a single unit. In religious and theological contexts, the related noun and adjective forms are more commonly encountered as Unitarier (members of the Unitarian movement) and Unitarismus (the doctrine), rather than unitär.

See also: unitary group, unitary matrix, unitary representation, unitarity.