endomorphism
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In abstract algebra, an endomorphism is a homomorphism from a mathematical object to itself. More generally in category theory, an endomorphism is a morphism from an object in some category to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space V is a linear map f: V → V, and an endomorphism of a group G is a group homomorphism f: G → G. frame|right|Orthogonal projection onto a line, , is a [[linear operator on the plane. This is an example of an endomorphism that is not an automorphism.]]
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Encyclopedic overview
8 sectionsContents
- Automorphisms
- Endomorphism rings
- Operator theory
- Endofunctions
- See also
- Notes
- References
- External links
In abstract algebra, an endomorphism is a homomorphism from a mathematical object to itself. More generally in category theory, an endomorphism is a morphism from an object in some category to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space V is a linear map f: V → V, and an endomorphism of a group G is a group homomorphism f: G → G. frame|right|Orthogonal projection onto a line, , is a [[linear operator on the plane. This is an example of an endomorphism that is not an automorphism.]]
In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set S to itself.
Excerpted from Wikipedia’s “endomorphism” article, available under the CC BY-SA 4.0 licence.