Invariant

In mathematics, the word invariant has according to the context various significances (nonequivalent). It is used as well in Géométrie and Topologie as in Analyze and Algèbre.

Invariant of a transformation

If F : E $\ rightarrow$ E is an application ensemblist, an invariant of F is an element X of E which is its own image under F , id is:

$f \left(X\right) =x$
In certain cases, the invariants of an application bring information about it.
• In Euclidean Geometry, the single invariant of a direct Similarity (which is not a translation) Euclidean plan will be its center.

• In reduction of the operators, a vectorial subspace F of E is known as invariant by a linear application has when the image of F under has is F . F can be carried out as a fixed point of the application induced by has on the whole of the vectorial subspaces of E .
• For an action (on the left) of a group G on a unit X , a point X is known as invariant when for any element G of G one a: G . X = X . The singleton X is an invariant of the transformation Y $\ mapsto G.Y$ induced by the action of G .

Under this significance, the term Point fixes is more usually used in dynamic Systèmes, for the geometrical transformations and the actions of group.

Invariant property

A property is known as invariant when a process does not modify it. A property relates to an object or a whole of objects given. Various constructions can be carried out to build objects of similar nature: part, complementary, nap, products, quotient, sticking together, extension,…

The invariance of a property characterizes its stability under these constructions.

Within the meaning of the theory of the categories

For a given category, an invariant is a quantity or an object associated (E) with each object with the category, and which depends only on the class of isomorphisms of the object, possibly except for isomorphism.

The language of the invariants is particularly adapted to the algebraic Topologie.

See too

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