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:

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".
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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.
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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 "ab".
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In mathematics, a quotient is the end result of a division problem. For example, in the problem 6 ÷ 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.
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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.
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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
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Topological equivalence redirects here; see also topological equivalence (dynamical systems).
In the mathematical field of topology, a homeomorphism or topological isomorphism
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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.
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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
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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.
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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.
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In mathematics, a complex number is a number of the form


where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number
(where and are real numbers) is



The complex conjugate is also very commonly denoted by .
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Topological equivalence redirects here; see also topological equivalence (dynamical systems).
In the mathematical field of topology, a homeomorphism or topological isomorphism
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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
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In matrix theory, a real 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


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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

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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.
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In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A
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In mathematics, the cross-ratio of a set of four distinct points on the complex plane is given by



This definition can be extended to the entire Riemann sphere (i.e. the complex plane plus the point at infinity) by continuity.
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In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A
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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.
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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.
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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.
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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.
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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.
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