Everything about Endomorphism totally explained
In
mathematics, an
endomorphism is a
morphism (or
homomorphism) from a mathematical object to itself. For example, an endomorphism of a
vector space V is a
linear map ƒ:
V →
V and an endomorphism of a
group G is a
group homomorphism ƒ:
G →
G, etc. In general, we can talk about endomorphisms in any
category. In the category of
sets, endomorphisms are simply functions from a set
S into itself.
In any category, the
composition of any two endomorphisms of
X is again an endomorphism of
X. It follows that the
set of all endomorphisms of
X forms a
monoid, denoted End(
X) (or End
C(
X) to emphasize the category
C).
An
invertible endomorphism of
X is called an
automorphism. The set of all automorphisms is a
subgroup of End(
X), called the
automorphism group of
X and denoted Aut(
X). In the following diagram, the arrows denote implication:
Any two endomorphisms of an
abelian group A can be added together by the rule (ƒ +
g)(
a) = ƒ(
a) +
g(
a). Under this addition, the endomorphisms of an abelian group form a
ring (the
endomorphism ring). For example, the set of endomorphisms of
Zn is the ring of all
n ×
n matrices with integer entries. The endomorphisms of a vector space,
module, ring, or
algebra also form a ring, as do the endomorphisms of any object in a
preadditive category. The endomorphisms of a nonabelian group generate an algebraic structure known as a
nearring.
Operator theory
In any
concrete category, especially for
vector spaces, endomorphisms are maps from a set into itself, and may be interpreted as
unary operators on that set,
acting on the elements, and allowing to define the notion of
orbits of elements, etc.
Depending on the additional structure defined for the category at hand (
topology,
metric, ...), such operators can have properties like
continuity,
boundedness, and so on.
More details should be found in the article about
operator theory.
Further Information
Get more info on 'Endomorphism'.
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