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In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class.

Invariants are used in diverse areas of mathematics such as geometry, topology and algebra. Some important classes of transformations are defined by an invariant they leave unchanged, for example conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important step in the process of classifying mathematical objects.

Simple examples

The most fundamental example of invariance is expressed in our ability to count. For a finite collection of objects of any kind, there appears to be a number to which we invariably arrive, regardless of how we count the objects in the set. The quantity—a cardinal number—is associated with the set, and is invariant under the process of counting.

An identity is an equation that remains true for all values of its variables. There are also inequalities that remain true when the values of their variables change.

Another 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.

Angles and ratios of distances are invariant under scalings, rotations, translations and reflections. These transformations produce similar shapes, which is the basis of trigonometry. All circles are similar. Therefore they can be transformed into each other and the ratio of the circumference to the diameter is invariant and equal to pi.

Some more complicated examples:

The real part and the absolute value of a complex number are invariant under complex conjugation.
The degree of a polynomial is invariant under linear change of variables.
The dimension and homology groups of a topological object are invariant 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).
Some invariants of projective transformations: collinearity of three or more points, concurrency of three or more lines, conic sections, the cross-ratio.
The determinant, trace, and eigenvectors and eigenvalues of a square matrix are invariant under changes of basis. In a word, the spectrum of a matrix is invariant to the change of basis.
Invariants of tensors.
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 integral $$\textstyle{\int_M K\,d\mu}$$ of the Gaussian curvature K of a 2-dimensional Riemannian manifold (M,g) is invariant under changes of the Riemannian metric g. This is the Gauss-Bonnet Theorem.

Invariant set

A subset S of the domain U of a mapping T: U → U is an invariant set under the mapping when x \in S \Rightarrow T(x) \in S. Note that the elements of S are not fixed, but rather the set S is fixed in the power set of U. (Some authors use the terminology setwise invariant vs. pointwise invariant to distinguish between these cases.) For example, a circle is an invariant subset of the plane under a rotation about the circle’s center. Further, a conical surface is invariant as a set under a homothety of space.

An invariant set of an operation T is also said to be stable under T. For example, the normal subgroups that are so important in group theory are those subgroups that are stable under the inner automorphisms of the ambient group. Other examples occur in linear algebra. Suppose a linear transformation T has an eigenvector v. Then the line through 0 and v is an invariant set under T. The eigenvectors span an invariant subspace which is stable under T.

When T is a screw displacement, the screw axis is an invariant line, though if the pitch is non-zero, T has no fixed points.

Formal statement

The notion of invariance is formalized in three different ways in mathematics: via group actions, presentations, and deformation.

Unchanged under group action

Firstly, if one has a group G acting on a mathematical object (or set of objects) X, then one may ask which points x are unchanged, "invariant" under the group action, or under an element g of the group.

Very frequently one will have a group acting on a set X and ask which objects in an associated set F(X) are invariant. For example, rotation in the plane about a point leaves the point about which it rotates invariant, while translation in the plane does not leave any points invariant, but does leave all lines parallel to the direction of translation invariant as lines. Formally, define the set of lines in the plane P as L(P); then a rigid motion of the plane takes lines to lines – the group of rigid motions acts on the set of lines – and one may ask which lines are unchanged by an action.

More importantly, one may define a function on a set, such as "radius of a circle in the plane" and then ask if this function is invariant under a group action, such as rigid motions.

Dual to the notion of invariants are coinvariants, also known as orbits, which formalizes the notion of congruence: objects which can be taken to each other by a group action. For example, under the group of rigid motions of the plane, the perimeter of a triangle is an invariant, while the set of triangles congruent to a given triangle is a coinvariant.

These are connected as follows: invariants are constant on coinvariants (for example, congruent triangles have the same perimeter), while two objects which agree in the value of one invariant may or may not be congruent (two triangles with the same perimeter need not be congruent). In classification problems, one seeks to find a complete set of invariants, such that if two objects have the same values for this set of invariants, they are congruent. For example, triangles such that all three sides are equal are congruent, via SSS congruence, and thus the length of all three sides forms a complete set of invariants for triangles.
Independent of presentation

Secondly, a function may be defined in terms of some presentation or decomposition of a mathematical object; for instance, the Euler characteristic of a cell complex is defined as the alternating sum of the number of cells in each dimension. One may forget the cell complex structure and look only at the underlying topological space (the manifold) – as different cell complexes give the same underlying manifold, one may ask if the function is independent of choice of presentation, in which case it is an intrinsically defined invariant. This is the case for the Euler characteristic, and a general method for defining and computing invariants is to define them for a given presentation and then show that they are independent of the choice of presentation. Note that there is no notion of a group action in this sense.

The most common examples are:

The presentation of a manifold in terms of coordinate charts – invariants must be unchanged under change of coordinates.
Various manifold decompositions, as discussed for Euler characteristic.
Invariants of a presentation of a group.

Unchanged under perturbation

Thirdly, if one is studying an object which varies in a family, as is common in algebraic geometry and differential geometry, one may ask if the property is unchanged under perturbation – if an object is constant on families or invariant under change of metric, for instance.

Erlangen program
Invariant (physics)
Invariant estimator in statistics
Invariant theory
Symmetry in mathematics
Topological invariant
Invariant differential operator
Invariant measure
Mathematical constant
Mathematical constants and functions

Notes

Fraleigh (1976, pp. 166–167)
Kay (1969, pp. 219)
Barry Simon. Representations of Finite and Compact Groups. American Mathematical Soc. p. 16. ISBN 978-0-8218-7196-6.
Judith Cederberg (1989). A Course in Modern Geometries. Springer. p. 174. ISBN 978-1-4757-3831-5.
Fraleigh (1976, p. 103)
Herstein (1964, p. 42)

McCoy (1968, p. 183)

References

Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0-201-01984-1
Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016
Kay, David C. (1969), College Geometry, New York: Holt, Rinehart and Winston, LCCN 69-12075
McCoy, Neal H. (1968), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68-15225
Weisstein, Eric W., "Invariant", MathWorld.
Popov, V.L. (2001), "Invariant", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4