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(c₁ v₁+c₂ v₂)= c₁ φ(v₁)+ c₂ φ(v₂)

for any c₁, c₂ in F and v₁, v₂ 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:

1. Associative law: χ_°(ψ_°φ) = (χ_°ψ)_°φ.
2. Distributive laws: (c₁ ψ₁+c₂ ψ₂)_°φ = c₁ ψ_1°φ + c₂ ψ_2°φ   and   ψ_°(c₁ φ₁ +c₂ φ₂) = c₁ ψ_°φ₁ + c₂  ψ_°φ₂, where c₁ and c₂ belong to F.
3. 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.