Invariant (mathematics) - Definition and Overview

An invariant in mathematics is something that does not change under a set of transformations. The property of being an invariant is invariance. For the laymen, let us just say an invariant is some kind of correspondence between two types of mathematical objects, so that two 'similar' things correspond to one and the same object. Invariants are useful in discriminating complicated objects.

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

Some examples, taking more complicated objects to numbers:

• 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 and under translations.
• The cross-ratio is invariant under projective transformations.
• The determinant and the trace 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.

See also

• fixed point (mathematics)
• invariant theory
• topological invariant