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Endomorphism

In mathematics, particularly within category theory, an endomorphism of an object is a morphism whose domain and codomain are the same object, effectively mapping the object onto itself while preserving its structure.^[1] This concept generalizes across various mathematical domains, where an endomorphism is a structure-preserving map (homomorphism) from an algebraic object to itself, such as a group to itself, a vector space to itself, or a module to itself.^[2] The set of all endomorphisms of a given object forms a monoid under composition of morphisms, with the identity morphism serving as the unit element.^[2] In abstract algebra, endomorphisms play a central role in studying symmetries and transformations within structures like groups, rings, and modules. For instance, the endomorphisms of an abelian group A constitute the endomorphism ring \operatorname{End}(A) = \operatorname{Hom}(A, A), where addition is pointwise and multiplication is given by composition, making it a ring with unity.^[3] In the context of vector spaces over a field, the endomorphisms are precisely the linear transformations from the space to itself, and for a finite-dimensional space of dimension n, the endomorphism ring is isomorphic to the ring of n \times n matrices over that field.^[2] A special case of endomorphisms are automorphisms, which are invertible endomorphisms, forming the automorphism group of the object.^[1] Endomorphisms are fundamental in advanced areas such as algebraic geometry and number theory; for example, the endomorphism ring of an elliptic curve over a field is either the integers \mathbb{Z} or an order in an imaginary quadratic field, influencing properties like the curve's arithmetic.^[4] They also arise in representation theory and functional analysis, where they describe self-maps on spaces or representations, often endowing them with additional algebraic structure.^[5]

General Concepts

Definition in Category Theory

In category theory, a category \mathcal{C} is a mathematical structure consisting of a class of objects and, for every pair of objects A and B, a set of morphisms (or arrows) from A to B, denoted \operatorname{Hom}_{\mathcal{C}}(A, B) or simply \mathcal{C}(A, B). These morphisms are equipped with a composition operation that is associative and has identity morphisms for each object, satisfying the axioms of a category.^[6] This framework abstracts the common patterns of structure-preserving maps across various mathematical domains, emphasizing relationships between objects via morphisms rather than the internal details of the objects themselves.^[6] An endomorphism of an object A in a category \mathcal{C} is formally defined as a morphism f: A \to A.^[7] The collection of all such endomorphisms forms a set, commonly denoted \operatorname{End}_{\mathcal{C}}(A) or simply \operatorname{End}(A) when the category is clear from context, which is precisely the hom-set \operatorname{Hom}_{\mathcal{C}}(A, A).^[6] This set captures all possible self-maps of A under the category's composition rule. The identity morphism \operatorname{id}_A: A \to A, which acts as the neutral element for composition, serves as the simplest example of an endomorphism in any category.^[7] More generally, if f: A \to A and g: A \to A are endomorphisms, their composition g \circ f: A \to A is also an endomorphism, endowing \operatorname{End}(A) with a monoid structure under composition, where the identity \operatorname{id}_A is the multiplicative unit and composition is associative by the category axioms.^[6] This monoid structure highlights the algebraic essence of endomorphisms as self-transformations that can be iteratively composed.

Basic Properties

Endomorphisms in a category are closed under composition: if f: A \to A and g: A \to A are endomorphisms of an object A, then their composite g \circ f: A \to A is also an endomorphism.^[7] This closure follows directly from the definition of composition in a category, where the codomain of f matches the domain of g.^[6] The composition of endomorphisms is associative, inheriting this property from the category axioms. Specifically, for any endomorphisms f, g, h: A \to A,

(g \circ f) \circ h = g \circ (f \circ h).

This associativity ensures that the operation is well-defined without ambiguity in the order of composition.^[6] The identity morphism \mathrm{id}_A: A \to A is itself an endomorphism and serves as the unit for composition: for any endomorphism f: A \to A, f \circ \mathrm{id}_A = \mathrm{id}_A \circ f = f.^[7] Consequently, the set \mathrm{End}(A) of all endomorphisms of A, equipped with composition as the binary operation and \mathrm{id}_A as the identity element, forms a monoid.^[6] This monoid structure, often denoted \mathrm{End}_C(A) when the category C needs specification, captures the algebraic properties of endomorphisms universally across categories.^[7] In categories with a zero object, such as abelian categories, there exists a zero morphism $0_{A,A}: A \to A for each object A, defined as the unique composite A \to 0 \to A through the zero object $0. This zero morphism is an endomorphism and acts as a zero element in the monoid \mathrm{End}(A), satisfying f \circ 0_{A,A} = 0_{A,A} \circ f = 0_{A,A} for any endomorphism f: A \to A.^[8]

Endomorphisms in Algebraic Structures

Groups and Monoids

In group theory, an endomorphism of a group G is a homomorphism \phi: G \to G that preserves the binary operation of the group. Specifically, for all a, b \in G, it satisfies the equation

\phi(ab) = \phi(a) \phi(b),

where the operation is written multiplicatively. This condition ensures that \phi respects the group structure, mapping the identity element to itself and preserving the associative law implicitly through the homomorphism property.^[9] A concrete example of a group endomorphism is the conjugation map induced by a fixed element g \in G, defined by \phi_g(x) = g x g^{-1} for all x \in G. This map preserves the group operation because

\phi_g(xy) = g (xy) g^{-1} = (g x g^{-1}) (g y g^{-1}),

making it a homomorphism from G to itself; moreover, it is bijective and thus an automorphism when the group is considered under this inner action.^[10] In additive abelian groups such as (\mathbb{Z}, +), endomorphisms include multiplication maps like \phi(n) = k n for a fixed integer k, which satisfy \phi(m + n) = k(m + n) = k m + k n = \phi(m) + \phi(n). For such endomorphisms \phi: G \to G, the kernel \ker(\phi) = \{ g \in G \mid \phi(g) = e \} forms a normal subgroup of G, since for any h \in G, \phi(h g h^{-1}) = \phi(h) \phi(g) \phi(h)^{-1} = e, implying h (\ker \phi) h^{-1} \subseteq \ker \phi. The image \operatorname{im}(\phi) = \{ \phi(g) \mid g \in G \} is a subgroup of G, as it is the homomorphic image under \phi.^[11] The concept extends naturally to monoids. An endomorphism of a monoid (M, \cdot, e) is a monoid homomorphism \psi: M \to M that preserves both the binary operation and the identity element, satisfying \psi(x y) = \psi(x) \psi(y) for all x, y \in M and \psi(e) = e. Unlike groups, monoids lack inverses, so their endomorphisms do not necessarily yield normal kernels, but the image remains a submonoid. The collection of all endomorphisms of a group G (or monoid M) under function composition forms the endomorphism monoid \operatorname{End}(G).^[12]^[13]

Rings and Modules

In ring theory, an endomorphism of a ring R is a ring homomorphism \phi: R \to R that preserves the ring operations. Specifically, it satisfies \phi(a + b) = \phi(a) + \phi(b) and \phi(ab) = \phi(a)\phi(b) for all a, b \in R, and typically \phi(1_R) = 1_R to ensure preservation of the multiplicative identity.^[14] These maps maintain the algebraic structure of the ring while mapping it to itself, distinguishing them from general homomorphisms to other rings. The set of all such endomorphisms, under pointwise addition and composition, forms the endomorphism ring \operatorname{End}_R(R).^[15] Common examples include the identity map \phi(a) = a, which is the unique endomorphism preserving every element, and the zero map \phi(a) = 0, though the latter fails to preserve the identity unless R is the zero ring. A notable non-trivial example is the Frobenius endomorphism on a field F of characteristic p > 0, defined by \phi(a) = a^p. This is a ring endomorphism because, in characteristic p, the freshman's dream holds: (a + b)^p = a^p + b^p and (ab)^p = a^p b^p, ensuring preservation of addition and multiplication. In finite fields, the Frobenius map is bijective, hence an automorphism. More generally, in perfect fields of characteristic p, the Frobenius endomorphism is bijective.^[14]^[16] For modules, an endomorphism of an R-module M is an R-linear map \phi: M \to M, meaning it is additive (\phi(m_1 + m_2) = \phi(m_1) + \phi(m_2)) and scalar-compatible (\phi(r m) = r \phi(m) for all r \in R, m \in M). The collection of all such maps forms the endomorphism ring \operatorname{End}_R(M), with addition defined pointwise and multiplication by composition. This linearity captures the module's action by the ring R, generalizing group endomorphisms to include scalar multiplication.^[15]^[17] A key property arises for free modules: if M is a free left R-module of finite rank n, then \operatorname{End}_R(M) \cong \operatorname{Mat}_n(R^{\operatorname{op}}), the ring of n \times n matrices over the opposite ring R^{\operatorname{op}}. When R is commutative, this simplifies to the matrix ring \operatorname{Mat}_n(R), where each endomorphism corresponds to left multiplication by a matrix with respect to a chosen basis. This correspondence facilitates computations, such as determinants for endomorphisms of free modules, and underscores the matrix representation's role in understanding module structure.^[18]^[19]

Linear Endomorphisms

Vector Spaces

In the context of vector spaces, an endomorphism is a linear map from a vector space to itself. Let V be a vector space over a field F. A linear endomorphism T: V \to V satisfies

T(\alpha \mathbf{u} + \beta \mathbf{v}) = \alpha T(\mathbf{u}) + \beta T(\mathbf{v})

for all \alpha, \beta \in F and \mathbf{u}, \mathbf{v} \in V.^[20] This definition applies to both finite- and infinite-dimensional vector spaces; in the infinite-dimensional case, endomorphisms are linear operators, and their study often incorporates topological or normed structures in functional analysis, though the algebraic properties remain foundational.^[21] For finite-dimensional vector spaces with \dim V = n < \infty, an ordered basis \{ \mathbf{e}_1, \dots, \mathbf{e}_n \} allows representation of T by an n \times n matrix A = (a_{ij}) over F, where the j-th column consists of the coordinates of T(\mathbf{e}_j) in the basis, so that if \mathbf{v} has coordinates \mathbf{x}, then T(\mathbf{v}) has coordinates A \mathbf{x}.^[22] A change of basis via an invertible matrix P yields a new representation B = P^{-1} A P; matrices related in this way are similar and represent the same endomorphism.^[23] The characteristic polynomial of T is

p_T(\lambda) = \det(\lambda I_n - A),

a monic polynomial of degree n that is independent of the basis choice, as similar matrices share the same characteristic polynomial.^[24] The minimal polynomial m_T(\lambda) is the monic polynomial of least degree such that m_T(T) = 0, dividing any annihilating polynomial of T (including p_T by the Cayley-Hamilton theorem) and determining the structure of the generalized eigenspaces.^[25] Over an algebraically closed field (such as \mathbb{C}), every finite-dimensional endomorphism T is similar to a unique (up to block permutation) Jordan canonical form J, a block-diagonal matrix with Jordan blocks

J_k(\lambda) = \begin{pmatrix} \lambda & 1 & & \\ & \lambda & \ddots & \\ & & \ddots & 1 \\ & & & \lambda \end{pmatrix}_{k \times k}

along the diagonal, where each block corresponds to an eigenvalue \lambda and the block sizes reflect the dimensions of the generalized eigenspaces.^[26] Key invariants under similarity include the trace \operatorname{tr}(T) = \sum_{i=1}^n a_{ii}, the sum of the diagonal entries (or eigenvalues with algebraic multiplicity), and the determinant \det(T) = \det(A), the product of the eigenvalues (with sign (-1)^n for the constant term of p_T). Both are preserved under basis changes, as \operatorname{tr}(P^{-1} A P) = \operatorname{tr}(A) and \det(P^{-1} A P) = \det(A).^[27]

Inner Product Spaces

In inner product spaces, linear endomorphisms gain additional structure through the inner product, allowing for notions like orthogonality and preservation of lengths. A Hilbert space H is a complete inner product space, and endomorphisms on H are linear operators T: H \to H. The inner product \langle \cdot, \cdot \rangle induces a norm \|u\| = \sqrt{\langle u, u \rangle}, enabling the study of operators that interact with this geometry.^[28] The adjoint operator T^* of a linear endomorphism T: H \to H is defined by the relation \langle T u, v \rangle = \langle u, [T^*](/page/Adjoint) v \rangle for all u, v \in H. This adjoint exists and is unique for bounded operators on Hilbert spaces, and it satisfies properties such as (T^*)^* = T and \|T\| = \|T^*\|. Self-adjoint endomorphisms, where T = T^*, have real eigenvalues and admit orthogonal diagonalization over the complex numbers; specifically, there exists an orthonormal basis of eigenvectors.^[29]^[30] Normal operators are those that commute with their adjoint, satisfying T T^* = T^* T. This class includes self-adjoint operators and generalizes them, preserving the spectral properties in finite dimensions. Unitary endomorphisms satisfy T^* T = I, where I is the identity, and thus preserve the inner product: \langle T u, T v \rangle = \langle u, v \rangle for all u, v \in H. More generally, isometries are endomorphisms satisfying \|T u\| = \|u\| for all u \in H, with unitary operators being surjective isometries.^[31]^[32] The spectral theorem for self-adjoint operators on finite-dimensional Hilbert spaces states that T = U D U^*, where U is unitary (i.e., U^* U = I) and D is diagonal with real entries on the diagonal, corresponding to the eigenvalues of T. This decomposition highlights the orthogonal structure inherent to inner products.^[33]

Endomorphism Rings

Ring Structure

The endomorphism ring of an object A in a category with biproducts, such as an abelian group, module, or vector space, is the set \operatorname{End}(A) of all endomorphisms of A, equipped with an addition operation defined pointwise and multiplication defined by composition of morphisms.^[5] For endomorphisms f, g \in \operatorname{End}(A), the sum is given by (f + g)(x) = f(x) + g(x) for every x \in A, where the addition on the right-hand side is the structure of A itself, and the product is (f \cdot g) = f \circ g, so that (f \circ g)(x) = f(g(x)).^[5] This makes \operatorname{End}(A) into an associative ring with unity.^[34] The multiplicative identity in \operatorname{End}(A) is the identity endomorphism \operatorname{id}_A, which satisfies \operatorname{id}_A \circ f = f \circ \operatorname{id}_A = f for all f \in \operatorname{End}(A).^[5] The additive identity is the zero morphism $0_A, defined by $0_A(x) = 0 for all x \in A, where $0 denotes the zero element in the additive structure of A.^[5] In general, \operatorname{End}(A) is a non-commutative ring, since composition of endomorphisms is not commutative except in trivial cases where all endomorphisms commute.^[5] For example, if f and g are distinct non-commuting endomorphisms, then f \circ g \neq g \circ f.^[34] The ring structure satisfies the distributive laws: for all f, g, h \in \operatorname{End}(A),

(f + g) \circ h = f \circ h + g \circ h, \quad h \circ (f + g) = h \circ f + h \circ g.

These follow from the pointwise definition of addition and the functoriality of composition with respect to addition in the category.^[5] For instance, the left distributivity holds because

((f + g) \circ h)(x) = (f + g)(h(x)) = f(h(x)) + g(h(x)) = (f \circ h)(x) + (g \circ h)(x)

for all x \in A.^[5] As a concrete illustration, the endomorphism ring of the n-dimensional vector space F^n over a field F is isomorphic to the ring of n \times n matrices over F.^[35]

Properties and Examples

In the endomorphism ring \operatorname{End}_R(A) of an R-module A, an idempotent is an element e \in \operatorname{End}_R(A) satisfying e \circ e = e, which corresponds to a projection onto the image e(A) along the kernel \ker(e).^[36] Such idempotents induce direct sum decompositions A = e(A) \oplus (1 - e)(A), where $1 denotes the identity endomorphism.^[37] The center Z(\operatorname{End}_R(A)) of the endomorphism ring consists of all elements that commute with every endomorphism in \operatorname{End}_R(A), forming a commutative subring.^[36] For a free R-module F, the center Z(\operatorname{End}_R(F)) is isomorphic to the center Z(R) of the base ring R.^[38] If \operatorname{End}_R(A) is a division ring, then A must be a simple R-module, as guaranteed by Schur's lemma, which states that the endomorphism ring of a simple module is a division ring.^[37] Conversely, simplicity of A ensures that every nonzero endomorphism is invertible.^[39] For an idempotent e \in \operatorname{End}_R(A), the Peirce decomposition of the endomorphism ring with respect to e decomposes the additive group as

\operatorname{End}_R(A) = e \operatorname{End}_R(A) e \oplus e \operatorname{End}_R(A) (1 - e) \oplus (1 - e) \operatorname{End}_R(A) e \oplus (1 - e) \operatorname{End}_R(A) (1 - e),

with multiplication defined by the ring structure connecting these components.^[40] Endomorphism rings play a central role in Morita equivalence, where two rings R and S are Morita equivalent if their module categories \operatorname{Mod}-R and \operatorname{Mod}-S are equivalent as categories; a key characterization is that S \cong \operatorname{End}_R(P) for some finitely generated projective generator P over R.^[36] Under Morita equivalence, the centers of the rings are isomorphic.^[41] Illustrative examples include the endomorphism ring \operatorname{End}_\mathbb{Z}(\mathbb{Z}) of the integers as a \mathbb{Z}-module, which is isomorphic to \mathbb{Z} itself via multiplication maps.^[36] In contrast, the endomorphism ring \operatorname{End}_k(k) of the polynomial ring k viewed as a module over a field k is far more complex, consisting of all k-linear maps on the infinite-dimensional vector space with basis \{1, x, x^2, \dots\} and forming a noncommutative ring without a simple closed form.^[36]

Automorphisms

Invertible Endomorphisms

An invertible endomorphism of an algebraic structure A is called an automorphism. It is a bijective endomorphism \phi: A \to A that admits an inverse \psi: A \to A satisfying \psi \circ \phi = \phi \circ \psi = \mathrm{id}_A.^[42]^[43] In the category of algebraic structures of the same type, automorphisms are precisely the isomorphisms from an object to itself.^[42] Automorphisms form a proper subset of the endomorphisms of A, consisting only of those that are bijective. Representative examples include permutation matrices, which induce automorphisms of the vector space \mathbb{R}^n by permuting the standard basis vectors.^[44] Another example is complex conjugation, which defines a field automorphism of \mathbb{C} by mapping a + bi \mapsto a - bi for a, b \in \mathbb{R}.^[45] The collection of all automorphisms of A, denoted \mathrm{Aut}(A), forms a group under function composition, with the identity morphism \mathrm{id}_A as the identity element. In the context of linear endomorphisms on a finite-dimensional vector space over a field, an endomorphism \phi is an automorphism if and only if its matrix representation has nonzero determinant, i.e., \det(\phi) \neq 0.

Automorphism Groups

For groups A, the automorphism group \operatorname{Aut}(A), consists of all automorphisms of A equipped with the group operation of composition; the identity map serves as the neutral element, and the inverse of any automorphism \phi is the map \phi^{-1} satisfying \phi \circ \phi^{-1} = \operatorname{id}_A.^[46] A distinguished normal subgroup of \operatorname{Aut}(A) is the inner automorphism group \operatorname{Inn}(A), generated by all conjugations \phi_g for g \in A, defined by the equation

\phi_g(h) = g h g^{-1}

for h \in A; this subgroup is normal because conjugation by elements of \operatorname{Aut}(A) preserves the form of inner automorphisms.^[47]^[48] Representative examples illustrate the structure of these groups. For the additive group \mathbb{Z} of integers, \operatorname{Aut}(\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}, generated by the inversion automorphism n \mapsto -n.^[49] For the symmetric group S_n with n \neq 6, \operatorname{Aut}(S_n) \cong S_n, so all automorphisms are inner; in contrast, S_6 admits outer automorphisms, making \operatorname{Aut}(S_6) a nontrivial extension of S_6.^[50] The outer automorphism group \operatorname{Out}(A) = \operatorname{Aut}(A)/\operatorname{Inn}(A) classifies automorphisms up to inner ones, capturing "essential" symmetries beyond conjugations.^[51] In finite groups, \operatorname{Aut}(G) is finite, with its order providing bounds on symmetries, such as |\operatorname{Aut}(G)| \leq |G|! in permutation representations.^[52] For infinite structures like the additive group of p-adic integers \mathbb{Z}_p, the automorphism group \operatorname{Aut}(\mathbb{Z}_p) \cong \mathbb{Z}_p^\times (the multiplicative units) is infinite and profinite, reflecting the topological completion in p-adic settings.^[53]

Endofunctions

Set Functions

In the category of sets, an endofunction on a set X is defined as any function f: X \to X, mapping every element of X to another element within the same set, without requiring any preservation of additional structure beyond the set-theoretic framework.^[54] This aligns with the general categorical notion of an endomorphism, where the endomorphisms of the object X in the category Set, denoted \mathrm{End}_{\mathrm{Set}}(X), coincide exactly with the collection of all such functions from X to itself. Representative examples of endofunctions include constant functions, which map every element of X to a fixed element c \in X, so f(x) = c for all x \in X; and permutations, which are the bijective endofunctions that rearrange the elements of X while preserving cardinality.^[55] Another key construction is iteration, where the n-fold composition f^{(n)}(x) applies f repeatedly n times to x, for positive integers n, allowing analysis of long-term behavior such as eventual periodicity in finite sets.^[56] The cardinality of the set of all endofunctions on X, denoted |\mathrm{End}(X)|, equals |X|^{|X|}, reflecting the independent choice of image for each element of X. For finite X with |X| = n, this simplifies to n^n.^[55] This exponential growth underscores the vast diversity of endofunctions even on small sets, with implications for enumerative combinatorics. A useful visualization of an endofunction f on X is its functional graph, a directed graph with vertex set X and a single directed edge x \to f(x) from each x \in X. The structure of such a graph decomposes into disjoint connected components, where each component features exactly one directed cycle, with directed trees feeding into the vertices of that cycle.^[56] This cycle-tree anatomy captures the dynamics of iteration, as repeated applications of f eventually lead elements into the cycles. Endofunctions under composition form a monoid, known as the full transformation monoid.^[56]

Monoids of Transformations

The set of all endofunctions from a set X to itself, often denoted \mathrm{End}(X) or T(X), forms a monoid under the operation of function composition, where the identity element is the identity function \mathrm{id}_X. This structure is associative because composition of functions is associative, and \mathrm{id}_X acts as a left and right identity for every endofunction. Without the identity, \mathrm{End}(X) is a semigroup, but the inclusion of \mathrm{id}_X elevates it to a monoid.^[57] For a finite set X = = \{1, \dots, n\}, the monoid T_n is the full transformation monoid, consisting of all n^n possible functions from

to itself. The group of units (invertible elements) in $T_n$ is the symmetric group $S_n$, which embeds as the submonoid of bijective transformations. A related structure is the monoid of partial transformations $\mathrm{PT}_n$, which includes functions with possibly undefined values on subsets of

and has cardinality (n+1)^n.^[58] Idempotent endofunctions in T(X) satisfy f \circ f = f, meaning applying f twice yields the same result as once. Such functions correspond bijectively to partitions of X: the kernel of f forms a partition of X into blocks, each of which is mapped uniformly to a distinct point in the image \mathrm{[im](/page/IM)}(f), with |\mathrm{[im](/page/IM)}(f)| equal to the number of blocks. For finite X = , the number of idempotents in T_n is \sum_{k=1}^n \binom{n}{k} k^{n-k}.^[59] The subsemigroup generated by these idempotents plays a key role in the structure of T_n.^[60]^[61] Green's relations provide a classification of elements in transformation semigroups like T_n. Specifically, the D-relation equates two transformations if they generate the same principal left and right ideals, and in T_n, the D-classes partition the monoid into sets of elements with the same rank, defined as \mathrm{rank}(f) = |\mathrm{im}(f)|. Thus, for each k = 1, \dots, n, there is a D-class D_k consisting of all transformations with image size k, and this rank is preserved under the relation. These classes reveal the ideal structure, with the constant maps (rank 1) forming the minimal ideal.^[62]

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[PDF] Chapter 1 Hilbert space and linear operators
Definition 1.1.1. A (complex) Hilbert space H is a vector space on C with a strictly positive scalar product (or inner product) which is complete for the ...
[29]
[PDF] functional analysis lecture notes: adjoints in hilbert spaces
A complex n × n matrix A is self-adjoint if and only if it is Hermitian, i.e., if. A = AH. Exercise 1.13. Show that every self-adjoint operator is normal. Show ...
[30]
[PDF] Adjoint operators - MTL 411: Functional Analysis
Self-adjoint, Normal and Unitary operators. Definition 1.4. A bounded linear operator T : H → H on a Hilbert space H is said to be. self-adjoint if T = T∗, ...
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[PDF] lecture 28: adjoints and normal operators
LECTURE 28: ADJOINTS AND NORMAL OPERATORS. Today's lecture will tie linear operators into our study of Hilbert spaces and discuss an important family of ...
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[PDF] Lecture 38: Unitary operators
If T : V −→ W , S : W −→ X are linear maps between inner product spaces preserving inner products, then S ◦ T also preserves inner products. Similary ...
[33]
[PDF] Proof of the spectral theorem
Nov 5, 2013 · Theorem 1.2 (Spectral theorem). Suppose V is a finite-dimensional real or complex vector space. The linear operator S ∈ L(V ) is selfadjoint if.
[34]
[PDF] On the arithmetic of the endomorphism ring End(Zp × Zpm) - arXiv
May 3, 2016 · Obviously, End(Zp × Zpm ) is a nonccommutative unitary ring with the usual addition and composition of endomorphisms. The proof in following ...
[35]
(PDF) Modules and their endomorphism rings - ResearchGate
Nov 30, 2019 · In fact, two vector spaces are isomorphic if and only if their endomorphism rings are isomorphic. Also in case module over a simple Artin ring ...
[36]
[PDF] An Introduction to Morita Theory - Matt Booth
In particular two commutative rings are Morita equivalent if and only if they are isomorphic. So Morita equivalence is only interesting for noncommutative rings ...
[37]
[PDF] math 101b: algebra ii part c: semisimplicity - Brandeis
Mar 22, 2007 · Theorem 1.10. The endomorphism ring EndR(M) of a simple module. M is a division ring. At this point I decided to review the definition ...
[38]
The center of a(n endomorphism) ring is a PID - MathOverflow
Apr 22, 2016 · For a ring R with identity and a free R-module F, the center of a EndR(F) is isomorphic (as ring) to the center of R. Here are two results from ...endomorphism rings of indecomposable objects - MathOverflowEndomorphism rings of infinitely generated free modules generated ...More results from mathoverflow.net
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For what modules is the endomorphism ring a division ring?
Nov 26, 2020 · Having a division ring as the endomorphism ring is equivalent to the condition that every non-zero endomorphism morphism is invertible.Galois theory of endomorphism rings of irreducible representationsLocal rings whose the endomorphism rings of E(R/J) is division ringMore results from mathoverflow.net
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[PDF] arXiv:1702.05261v1 [math.RA] 17 Feb 2017
Feb 17, 2017 · Abstract. Idempotents dominate the structure theory of rings. The Peirce decomposition induced by an idempotent provides a natural environment ...
[41]
[PDF] Morita Equivalence - Alistair Savage
If two rings are Morita equivalent, then their centers are isomorphic. In particular, if the rings are commutative, then they are isomorphic. Because of the ...
[42]
Morphism -- from Wolfram MathWorld
A surjective morphism from an object to itself is called an endomorphism, and. 6. An isomorphism between an object and itself is called an automorphism. See ...
[43]
homomorphism between algebraic systems - PlanetMath.org
Mar 22, 2013 · finally, an automorphism is an endomorphism that is also an isomorphism. •. All trivial algebraic systems (of the same type) are isomorphic. •.
[44]
[PDF] Lecture 2. The Categories of Vector Spaces
Permutation Matrices. Consider fτ : kn → kn that permutes the ... The dual of the vector space V is the vector space: V ∨ = {linear functions ...
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Field Automorphism -- from Wolfram MathWorld
A field automorphism of a field is a bijective map that preserves all of 's algebraic properties, more precisely, it is an isomorphism. For example, complex ...
[46]
[PDF] Lecture 4.6: Automorphisms - Mathematical and Statistical Sciences
The automorphism group of Zn is Aut(Zn) = {σa | a ∈ U(n)} ∼= U(n), where σa : Zn −→ Zn , σa(1) = a . M. Macauley (Clemson). Lecture 4.6: Automorphisms. Math ...
[47]
[PDF] Automorphisms
Conjugation by g is called an inner automorphism of G and the subgroup of Aut(G) consisting of all inner automorphisms is denoted Inn(G). Kevin James.
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[PDF] 19. Automorphism group of Sn - UCSD Math
Then the inner automorphism group is a normal subgroup of A(G). The quotient group Aut(G)/Inn(G) is called the outer automor- phism group of G, denoted Out(G).
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[PDF] Math 5863 homework solutions 31. (3/22) Denote the automorphism ...
The choice φ(1) = −1 gives φ(n) = −n for all n, that is, φ is multiplication by −1. So Aut(Z) is a group with exactly two elements, hence Aut(Z). ∼. = C2.
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[PDF] 9 Automorphism groups - Brandeis
The automorphism group (Aut(G)) of a group G is the set of automorphisms of G, which form a group under composition. Inner automorphisms are part of Aut(G).
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[PDF] Chapter 4: Maps between groups
An outer automorphism of G is any automorphism that is not inner. The outer automorphism group of G is the quotient Out(G) := Aut(G)/ Inn(G). Aut(D4).
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[PDF] About Automorphisms of Some Finite Groups
The set of inner automorphisms of a group G is a normal subgroup of the group of ... groups, having trivial centers, are isomorphic to their inner automorphism ...
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[PDF] Analytic Pro-p groups (Second Edition) - School of Arts & Sciences
Automorphism groups. The group GLd(Zp). The automorphism group of a profinite group. Automorphism groups of pro-p groups. Finite extensions. Exercises.
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[PDF] endofunctions of given cycle type harald fripertinger and peter sch ¨opf
An endofunction on the set X is a function f with domain and range X. The term endofunction comes from species theory. See for instance [2, 3, 4, 5, 10].
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[PDF] Naive set theory. - Whitman People
A more important way in which the naive point of view predominates is that set theory is regarded as a body of facts, of.
[56]
[PDF] Minimal representatives of endofunctions
Since endofunctions are in one-to-one correspondence with functional digraphs, our prob- lem can be solved by graph algorithms. It is known that canonical ...
[57]
[PDF] Permutation Groups and Transformation Semigroups Lecture 2
Almost the same is true for semigroups: Proposition 4.1 Any semigroup of order n is isomorphic to a subsemigroup of the full transformation semigroup Tn+1.Missing: T_n | Show results with:T_n
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[PDF] The ranks of ideals in various transformation monoids
Sep 13, 2013 · As usual we denote by PTn the monoid of all partial transformations of a finite set Xn with n elements (under composition), by Tn the ...Missing: T_n | Show results with:T_n
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Idempotent generation in the endomorphism monoid of a uniform ...
Jul 11, 2014 · Idempotent generation in the endomorphism monoid of a uniform partition. Denote by \mathcal T_n and \mathcal S_n the full transformation ...Missing: endofunctions | Show results with:endofunctions
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Idempotent Generation in the Endomorphism Monoid of a Uniform ...
Jul 6, 2016 · We enumerate the idempotents of 𝒯(X, 𝒫), and describe the submonoid S = ⟨ E ⟩ generated by the idempotents E = E(𝒯(X, 𝒫)).Missing: endofunctions | Show results with:endofunctions
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Regularity and Green′s Relations on a Semigroup of ...
Nov 5, 2008 · Finally, we define ℋ = ℒ∩ℛ and D = ℒ∘ℛ. In [2, 3], Clifford and Preston characterized Green′s relations on the full transformation semigroup T(X) ...