In mathematics, an invariant is something that does not change under a set of transformations. The property of being an invariant is invariance. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a transformation in elementary terms is any of a variety of different operations from geometry, such as rotations, reflections and translations. ...

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

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: A number line is a one-dimensional graph. ... Addition (or summation) is one of the basic operations of arithmetic. ... In its simplest form, multiplication is a quick way of adding identical numbers. ...

• The real numbers 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.
• 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.

In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, a fixed point of a function is a point that is mapped to itself by the function. ... A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ... In linear algebra, an orthogonal matrix is a square matrix G whose transpose is its inverse, i. ... In mathematics, the cross-ratio cr( w, x, y, z ) of an ordered quadruple of complex numbers (which may be real numbers) is Cross-ratios are preserved by linear fractional transformations, i. ... In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... In linear algebra the singular value decomposition (SVD) is a factorization of a rectangular real or complex matrix analogous to the diagonalization of symmetric or Hermitian square matrices using a basis of eigenvectors (see spectral theorem). ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...

See also

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