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:
- Associative law: χ°(ψ°φ) = (χ°ψ)°φ.
- Distributive laws: (c1 ψ1+c2 ψ2)°φ = c1 ψ1°φ + c2 ψ2°φ and ψ°(c1 φ1 +c2 φ2) = c1 ψ°φ1 + c2 ψ°φ2, where c1 and c2 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 EndF(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 EndF(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.