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Many linear maps compare one [vector space](/page/Vector%20Space) with another, but some of the richest questions begin when the source and target are the same. Then a map can be iterated, represented by a square matrix, tested for fixed points and eigenvectors, and inserted into polynomials. This is why finite-dimensional linear algebra treats maps $T: V \to V$ as objects in their own right rather than merely as arrows between spaces.

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For a finite-dimensional vector space $V$ over a field $F$, these self-maps form the endomorphism algebra $\operatorname{End}_F(V)$. It packages projections, nilpotent operators, changes of basis, kernels, images, and matrix similarity into one setting. The same idea also appears more broadly in [Function](/page/Function), [Linear Map](/page/Linear%20Map), [Group Homomorphism](/page/Group%20Homomorphism), and Ring Homomorphism, but the linear finite-dimensional case is the guiding model on this page.

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## Definition

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In undergraduate linear algebra, the object being studied is usually a finite-dimensional vector space, and the structure to preserve is linear structure. Requiring the codomain to be the same vector space is what makes powers $T^m$, eigenvectors, characteristic polynomials, and matrix similarity available.

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[definition: Endomorphism] Let $V$ be a finite-dimensional vector space over a field $F$. An endomorphism of $V$ is a linear map \begin{align*} T: V &\to V. \end{align*} The set of all endomorphisms of $V$ is denoted $\operatorname{End}_F(V)$. [/definition]

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## Variants

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The finite-dimensional hypothesis is not needed to define a linear self-map, but it is the setting where every endomorphism can be represented by a square matrix after choosing a basis. This distinction motivates the following definition, which keeps the self-map condition while dropping any unnecessary finite-dimensional assumption.

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[definition: Linear Endomorphism] Let $V$ be a vector space over a field $F$. A linear endomorphism of $V$ is a linear map \begin{align*} T: V &\to V. \end{align*} The set of all linear endomorphisms of $V$ is denoted $\operatorname{End}_F(V)$ or $\mathcal{L}(V,V)$. [/definition]

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The word "endomorphism" also has a categorical form, useful when the same pattern is repeated for groups, rings, modules, and topological spaces. This broader version is needed to state the common pattern once: the ambient category tells us which structure the self-map must preserve.

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[definition: Categorical Endomorphism] Let $\mathcal{C}$ be a category and let $X$ be an object of $\mathcal{C}$. A categorical endomorphism of $X$ is a morphism \begin{align*} f: X &\to X \end{align*} in $\mathcal{C}$. The set of endomorphisms of $X$ is denoted $\operatorname{End}_{\mathcal{C}}(X)$. [/definition]

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The linear condition means that for all $v,w \in V$ and all $a \in F$,

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\begin{align*} T(v+w) &= T(v)+T(w), & T(av) &= aT(v). \end{align*}

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Because the codomain is again $V$, iterates such as $T^2=T\circ T$ and $T^3=T\circ T\circ T$ can be formed. The later algebraic structure theorem and polynomial-closure theorem state the formal closure results that justify building new endomorphisms from these iterates.

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For groups, an arbitrary self-map need not respect the multiplication, so iterating it can destroy the algebraic information one wants to study. The useful self-maps are those compatible with the group law, because their powers, kernels, quotients, and induced actions remain tied to the same group structure.

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[definition: Group Endomorphism] Let $G$ be a group. A group endomorphism of $G$ is a group homomorphism \begin{align*} \varphi: G &\to G. \end{align*} [/definition]

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Rings require a separate self-map notion because they carry two interacting operations. Preserving addition alone is not enough: an additive self-map can fail to respect products, so its iterates may no longer reflect the multiplication that makes the ring a ring.

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The obstruction is that many natural additive operators, such as multiplication by a fixed element, are self-maps of the underlying [abelian group](/page/Abelian%20Group) without being compatible with multiplication in the ring. To study symmetries, dynamics, or repeated operations internal to a ring, one needs a self-map whose powers still preserve both ring operations. The next definition isolates exactly those self-maps: the map must be a homomorphism of the ring back to itself.

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[definition: Ring Endomorphism] Let $R$ be a ring. A ring endomorphism of $R$ is a ring homomorphism \begin{align*} \psi: R &\to R. \end{align*} [/definition]

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The convention about whether ring homomorphisms preserve the multiplicative identity is inherited from the ambient category of rings being used. This is another instance of the same warning: an endomorphism is always a self-map preserving the structure that has been specified.

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## Algebraic Structure on Endomorphisms

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