Definition
In mathematics, an invariant is something that does not change under a set of transformations. The property of being an invariant is invariance. 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 <math>\sim</math> on it, an invariant is a function <math>f\colon X \to Y</math> that is constant on equivalence classes: it doesn't depend on the particular element. Equivalently, it descends to a function on the quotient <math>X/\sim</math>. 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 <math>X</math> is called forward invariant under <math>f</math> if <math>f(X)= X</math>, and backward invariant if <math>f^{-1}(X) = X</math>. A set is completely invariant under <math>f</math> if it is both forward and backward invariant under <math>f</math>.
Examples
One 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, under complex conjugation.
- The degree of a polynomial, under linear change of variables.
- The dimension of a topological object, 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.
- The singular values of a matrix are invariant under orthogonal transformations.
- 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 the complex function <math>f</math> is completely invariant under <math>f</math>
