Invariantie

Invariantie, or invariance, is a property that remains unchanged under a specified class of transformations. In mathematics, physics and computer science, invariants help identify essential features, reveal symmetry, and support reasoning about objects and processes.

Formally, if a transformation group G acts on a set X and a quantity f: X → Y is defined on X, then f is invariant under G if f(g·x) = f(x) for all g in G and x in X. In computing, invariants also appear as loop invariants—conditions that hold before and after each iteration of a loop, assisting in proving algorithm correctness.

Common geometric invariants include distances and lengths under isometries like rotations and reflections, and angles under similarity transformations. In linear algebra, the determinant is invariant under similarity transforms: det(PAP⁻¹) = det(A). In physics, conserved quantities such as energy and momentum are invariants under the system’s evolution, and the spacetime interval is invariant under Lorentz transformations in special relativity.

In computer science, invariants guide program analysis and verification. A loop invariant might ensure that a data structure remains correctly ordered or that a running total matches the sum of processed elements at every step. In data processing and cryptography, invariants help characterize objects and detect errors or tampering under transformations.

Applications span geometry, algebra, dynamical systems, physics, and formal verification. The study of invariants is closely linked to symmetry and group theory, with Noether’s theorem connecting symmetries to conserved quantities.

See also: invariants, symmetry, Noether’s theorem, loop invariant, conserved quantity, group action.