# invariant (mathematics)

## 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 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 is called

**forward invariant**under if , and

**backward invariant**if . A set is

**completely invariant**under if it is both forward and backward invariant under .

## 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 is completely invariant under

## See also

**Mathematics**(colloquially,

**maths**or

**math**) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".

**.....**Click the link for more information.

In mathematics, a

**transformation**in elementary terms is any of a variety of different functions from geometry, such as rotations, reflections and translations. These can be carried out in Euclidean space, particularly in dimensions 2 and 3.**.....**Click the link for more information. In mathematics, an

**equivalence relation**is a binary relation between two elements of a set which groups them together as being "equivalent" in some way. That*a*is equivalent to*b*is denoted as "*a*~*b*" or "*a*≡*b*".**.....**Click the link for more information. In mathematics, a

**quotient**is the end result of a division problem. For example, in the problem 6 Ã· 3, the quotient would be 2, while 6 would be called the dividend, and 3 the divisor. The quotient can also be expressed as the number of times the divisor divides into the dividend.**.....**Click the link for more information. In mathematics,

**category theory**deals in an abstract way with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and have been a unifying notion.**.....**Click the link for more information.**functor**is a special type of mapping between categories. Functors can be thought of as morphisms in the category of small categories.

Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological

**.....**Click the link for more information.

Not to be confused with homomorphism.

*Topological equivalence**redirects here; see also topological equivalence (dynamical systems).*

**homeomorphism**or

**topological isomorphism**

**.....**Click the link for more information.

For a general, non-technical overview of the subject, see .

**Topological spaces**are mathematical structures that allow the formalization of concepts such as convergence, connectedness and continuity.

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**Complex analysis**, traditionally known as the

**theory of functions of a complex variable**, is the branch of mathematics investigating functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics.

**.....**Click the link for more information.

A

**number line**, invented by John Wallis, is a one-dimensional picture in which the integers are shown as specially-marked points evenly spaced on a line. Although this image only shows the integers from -9 to 9, the line includes all real numbers, continuing "forever" in each**.....**Click the link for more information.**Addition**is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. Addition also provides a model for related processes such as joining two collections of objects into one collection.

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**Multiplication**is the mathematical operation of adding together multiple copies of the same number. For example, four multiplied by three is twelve, since three sets of four make twelve:

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**real part**of a complex number , is the first element of the ordered pair of real numbers representing , i.e. if , or equivalently, , then the real part of is . It is denoted by Re or , where is a capital R in the Fraktur typeface.

**.....**Click the link for more information.

In mathematics, the

**absolute value**(or**modulus**^{[1]}) of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.**.....**Click the link for more information. In mathematics, a

where

**complex number**is a number of the formwhere

*a*and*b*are real numbers, and*i*is the imaginary unit, with the property*i*Â² = −1.**.....**Click the link for more information.**complex conjugate**of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number

The complex conjugate is also very commonly denoted by .

**.....**Click the link for more information.

Not to be confused with homomorphism.

*Topological equivalence**redirects here; see also topological equivalence (dynamical systems).*

**homeomorphism**or

**topological isomorphism**

**.....**Click the link for more information.

**fixed point**(sometimes shortened to

**fixpoint**) of a function is a point that is mapped to itself by the function. That is to say, is a fixed point of the function if and only if .

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**dynamical system**concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and

**.....**Click the link for more information.

In matrix theory, a real

An orthogonal matrix is a

**orthogonal matrix**is a square matrix`Q`whose transpose is its inverse:An orthogonal matrix is a

**special orthogonal matrix**if it has determinant +1:## Overview

**.....**Click the link for more information.**Area**is a physical quantity expressing the size of a part of a surface. The term

**Surface area**is the summation of the areas of the exposed sides of an object.

### Units

Units for**measuring surface area**include:

- square metre = SI derived unit

**.....**Click the link for more information.

In mathematics, a

**linear map**(also called a**linear transformation**or**linear operator**) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.**.....**Click the link for more information. In algebra, a

**determinant**is a function depending on*n*that associates a scalar, det(*A*), to every*n*Ã—*n*square matrix*A*. The fundamental geometric meaning of a determinant is as the scale factor for volume when*A***.....**Click the link for more information. In mathematics, the

This definition can be extended to the entire Riemann sphere (i.e. the complex plane plus the point at infinity) by continuity.

**cross-ratio**of a set of four distinct points on the complex plane is given byThis definition can be extended to the entire Riemann sphere (i.e. the complex plane plus the point at infinity) by continuity.

**.....**Click the link for more information. In algebra, a

**determinant**is a function depending on*n*that associates a scalar, det(*A*), to every*n*Ã—*n*square matrix*A*. The fundamental geometric meaning of a determinant is as the scale factor for volume when*A***.....**Click the link for more information. 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.e.**.....**Click the link for more information.**eigenvector**of the transformation and the blue vector is not. Since the red vector was neither stretched nor compressed, its

**eigenvalue**is 1. All vectors with the same vertical direction - i.e., parallel to this vector - are also eigenvectors, with the same eigenvalue.

**.....**Click the link for more information.

**eigenvector**of the transformation and the blue vector is not. Since the red vector was neither stretched nor compressed, its

**eigenvalue**is 1. All vectors with the same vertical direction - i.e., parallel to this vector - are also eigenvectors, with the same eigenvalue.

**.....**Click the link for more information.

In linear algebra, the

**singular value decomposition**(**SVD**) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics.**.....**Click the link for more information. In mathematics, the

**Lebesgue measure**, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration.**.....**Click the link for more information.This article is copied from an article on Wikipedia.org - the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.