Invariants

An invariant is a property of a mathematical object that remains unchanged under a specified group of transformations or operations. Formally, if a set X is acted on by a group G, a quantity f: X → Y is invariant under G if f(g·x) = f(x) for all g ∈ G and x ∈ X. Invariants are central to classifying objects up to the corresponding notion of equivalence.

Common examples include: distances and angles are invariant under rigid motions (translations, rotations, reflections) of the plane or space. The Euclidean norm of a vector is invariant under orthogonal transformations. Under similarity transformations, determinants and traces of a matrix are invariant: det(P^{-1}AP) = det(A) and tr(P^{-1}AP) = tr(A). Eigenvalues are invariant under similarity. In topology, the Euler characteristic is a topological invariant preserved by homeomorphisms. In linear algebra, many properties such as the spectrum are preserved under similarity.

In computer science and discrete mathematics, invariants appear in proofs and algorithms. Loop invariants are conditions that hold before and after each iteration of a loop and are used to establish correctness. In physics, conserved quantities such as energy, momentum, and electric charge are invariants under the time evolution of a closed system. In geometry and topology, invariants such as genus, homology groups, and other algebraic topological data distinguish spaces up to homeomorphism or diffeomorphism, and graph invariants like the chromatic number remain unchanged under graph isomorphism.

Overall, invariants capture what remains fixed under a given set of transformations, providing a unifying tool for comparison, classification, and rigorous reasoning.