# Endomorphism

In linear algebra, an **endomorphism** is a linear mapping φ of a linear space *V* into itself, where *V* is assumed to be over the field of numbers **F**. (Outside of pure mathematics **F** is usually either the field of real or complex numbers).
A mapping φ is linear if

- φ(
*c*_{1}*v*_{1}+*c*_{2}*v*_{2})=*c*_{1}φ(*v*_{1})+*c*_{2}φ(*v*_{2})

for any *c*_{1}, *c*_{2} in **F** and *v*_{1}, *v*_{2} in *V*.

Since the product of two endomorphisms φ and ψ is again an endomorphism of *V*, the multiplication φ_{°}ψ associates with any two endomorphisms of *V* a third endomorphism. This multiplication has the following properties:

- Associative law: χ
_{°}(ψ_{°}φ) = (χ_{°}ψ)_{°}φ. - Distributive laws: (
*c*_{1}ψ_{1}+*c*_{2}ψ_{2})_{°}φ =*c*_{1}ψ_{1}_{°}φ +*c*_{2}ψ_{2}_{°}φ and ψ_{°}(*c*_{1}φ_{1}+*c*_{2}φ_{2}) =*c*_{1}ψ_{°}φ_{1}+*c*_{2}ψ_{°}φ_{2}, where*c*_{1}and*c*_{2}belong to**F**. - There exists an endomorphism ι (the identity map) such that φ
_{°}ι = ι_{°}φ = φ for every endomorphism φ.

Note that the product is not commutative, i.e., in general ψ_{°}φ ≠φ_{°}ψ.
The set of all endomorphisms forms an associative algebra. That is, the set is a linear space with multiplication. This algebra is often denoted by End_{F}(*V*) or by L(*V*,*V*).

An endomorphism φ is called an **automorphism** of *V* if it is invertible, that is, there exists an endomorphism φ^{−1} such that φ^{−1}_{°}φ = φ_{°}φ^{−1} = ι. The set of automorphisms is not a linear space (a linear combination of automorphisms is not generally invertible). The set is, however, a group, denoted by GL(*V*, **F**) (the general linear group of automorphisms on *V*).

If *V* is finite dimensional, say of dimension *n*, the algebra End_{F}(*V*) is algebra-isomorphic with the algebra of *n*×*n* matrices with elements in **F**. The element ι is in 1-1 correspondence with the *n*×*n* identity matrix **I**. The elements of GL(*V*, **F**) are in 1-1 correspondence with non-singular (invertible) matrices.