Endomorphism

In universal algebra is an endomorphism (from the Greek endo ἔνδον inside and Greek μορφή morph shape, form ) is a homomorphism of a mathematical structure in itself is also an isomorphism, then automorphism is called.

In category theory ie, each morphism whose source and target match, an endomorphism of the object in question.

The totality of the endomorphism of an object A is indicated by end (A) and always forms a monoid ( the Endomorphismenmonoid or Endomorphismenhalbgruppe ) in additive categories even a ( unitary ) ring.

• 2.1 Vector Spaces 2.1.1 General
• 2.1.2 example

Definition

Algebraic Structures

Be an algebraic structure, so a lot together with a finite number of links with appropriate arities. Such an algebraic structure could for example be a vector space, a group or a ring. Then one sees in the algebra under a linear operator a mapping of the set onto itself, which is a homomorphism, ie it is

For everyone.

Category theory

Be an object of a category. A morphism which operates an object on the is called endomorphism.

For categories of homomorphisms between algebraic structures is the definition equivalent to that in the previous section.

Special Structures

Vector spaces

General

In linear algebra, an endomorphism or Vektorraumendomorphismus is a linear map. This is referred to as a K- vector space and a linear map meant a figure that

For all and all fulfilled. Together with the composition as multiplication the set of all endomorphisms of a ring, which is called the endomorphism ring. If the underlying vector space is a topological vector space, and considers the vector space of continuous linear operators, which is a real subspace of Endomorphismenraums in the case of infinite-dimensional vector spaces in general, one can induce this vector space of all continuous endomorphisms of a topology such that addition and multiplication are the ring constantly. Thus, the endomorphism ring is a topological ring.

Example

The differential is on the vector space of polynomials up to the third degree with real coefficients is an endomorphism. As a basis of V is chosen, the monomial basis. These can be isomorphic to the canonical base map of, by. The 1 stands at the i -th digit of the 4- tuple. Thus, one can from each polynomial represented as 4- tuples, then for example. Now you can link with and receives a matrix notation for the differential:

Applying this matrix to the above example, suppose we obtain, corresponding to the polynomial; which could have been obtained by directly applying the derivative.

Groups

An endomorphism of a group is a group homomorphism from to, that is for valid for all.