Invariant

In mathematics, an invariant is something that does not change under a set of transformations. The property of being an invariant is invariance.

However, it may be more accurate to say that invariance is the idea that a system maintains the correlation of its parameters when undergoing a transformation. Invariance mathematically relates to perceptual studies of the visual field and has been questioned in regard to its quality.

Mathematicians say that a quantity is invariant "under" a transformation; some economists say it is invariant "to" a transformation.

More generally, given a set X with an equivalence relation ˜ on it, an invariant is a function that is constant on equivalence classes: it doesn't depend on the particular element. Equivalently, it descends to a function on the quotient .

The transform definition of invariant is a special case of this, where the equivalence relation is "there is a transform that takes one to the other".

In category theory, one takes objects up to isomorphism; every functor defines an invariant, but not every invariant is functorial (for instance, the center of a group is not functorial).

In computational approaches to math, one takes presentations of objects up to isomorphism, such as presentations of groups or simplicial sets up to homeomorphism of the underlying topological space.

In complex analysis, set X is called forward invariant under f if f(X) = X, and backward invariant if f ^{− 1}(X) = X. A set is completely invariant under f if it is both forward and backward invariant under f.

Examples

The most fundamental example of invariance is expressed in our ability to count. For a finite collection of objects of any kind, there appears to be a number to which we invariably arrive regardless of how we count the objects in the set. The quantity – a cardinal number – is associated with the set and is invariant under the process of counting.

Another simple example of invariance is that the distance between two points on a number line is not changed by adding the same quantity to both numbers. On the other hand multiplication does not have this property so distance is not invariant under multiplication.

Some more complicated examples:

• The real part and the absolute value of a complex number are invariant under complex conjugation.
• The degree of a polynomial is invariant under linear change of variables.
• The dimension of a topological object is invariant under homeomorphism.
• The number of fixed points of a dynamical system is invariant under many mathematical operations.
• Euclidean distance is invariant under orthogonal transformations.
• Euclidean area is invariant under a linear map with determinant 1 (see Equi-areal maps).
• The cross-ratio is invariant under projective transformations.
• The determinant, trace, and eigenvectors and eigenvalues of a square matrix are invariant under changes of basis.
• Invariants of tensors.
• The singular values of a matrix are invariant under orthogonal transformations.
• The Midline Theorem is an invariant, where line xy is 1/2 of base bc.
• Lebesgue measure is invariant under translations.
• The variance of a probability distribution is invariant under translations of the real line; hence the variance of a random variable is unchanged by the addition of a constant to it.
• The fixed points of a transformation are the elements in the domain invariant under the transformation. They may, depending on the application, be called symmetric with respect to that transformation. For example, objects with translational symmetry are invariant under certain translations.
• The Fatou set and Julia set generated by a complex function f are completely invariant under f.
• The integral

of the Gaussian curvature K of a 2-dimensional Riemannian manifold (M,g) is invariant under changes of the Riemannian metric g. This is the Gauss-Bonnet Theorem.

• More generally, characteristic classes are invariants of smooth manifolds M, under diffeomorphism. For example, a smooth manifold is orientable if, and only if, its first Stiefel-Whitney class vanishes. A smooth orientable manifold is spin if, and only if, its second Stiefel-Whitney class vanishes.

See also

• Erlangen program
• invariant in physics
• Invariant estimator in statistics
• Invariant theory
• Symmetry in mathematics
• topological invariant
• Invariant differential operator

External links

• The Beginning of Mathematics from cut-the-knot