Ainvariant

Ainvariant, usually written as A-invariant, is a term used to describe a property that remains unchanged under the action of a specified operator, transformation, or group. In mathematics it is a general notion of invariance; the prefix A helps specify the particular transformation or family of transformations being considered.

In linear algebra, an A-invariant subspace W of a vector space V is one in which applying A to any vector in W yields another vector in W. That is, A(W) is contained in W. Invariant subspaces are fundamental to analyzing linear operators, enabling decompositions, eigenstructure studies, and the characterization of matrix forms such as Jordan form. For many operators, the span of eigenvectors corresponding to a set of eigenvalues forms an invariant subspace.

In algorithm design and computer science, a loop invariant is a condition that remains true before and after every iteration of a loop. Loop invariants are used to reason about and prove the correctness of algorithms. For example, a sorting loop might maintain an invariant that a certain prefix of the array is sorted or that elements satisfy a particular ordering property at each step.

In group theory and geometry, invariants refer to properties that remain unchanged under a group of symmetries or transformations. Examples include fixed points, curvature, or algebraic invariants that classify objects up to symmetry. Invariant theory studies these preserved quantities under group actions and their consequences in algebra and geometry.

See also invariants, invariant subspace, loop invariant, and invariant theory for related concepts and applications.