Endomorphism

Endomorphism
Tag	`0PTF1PTJK0`
URI	`m2://0PTF1PTJK0`
Kind	definition
Writing	W0 stub
MSC	15A04
Defines	Endomorphism
Uses	Linear map

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Definition

Let $F$ be a field and let $V$ be a vector space over $F$ . An endomorphism of $V$ is a linear map $T \colon V \to V$ . The set of all endomorphisms of $V$ is written $\End_F(V)$ .

Remarks

An endomorphism is also called a linear operator on $V$ . Endomorphisms can be composed with each other, so their powers $T^k$ and polynomials $p(T)$ are defined. Under composition and addition, $\End_F(V)$ is the algebra of endomorphisms. Eigenvalues, the trace and the determinant are defined for endomorphisms of finite-dimensional spaces.

Explanations

L1 human

An endomorphism maps a space into itself. So we can apply it again and again, and compare a vector with its image. In a basis, an endomorphism of an $n$ -dimensional space is a square $n \times n$ matrix.

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Vector space definition
Linear map definition
Endomorphism definition

What depends on this

53 nodes use this node, directly or through other nodes.

Singular value decomposition 000NPTMX6J
Jordan–Chevalley decomposition 011K11VPJN
Reflection 025158VEYZ
Determinant of a linear map 03F5KP94K0
Spectrum of a matrix 04EHB7YDQS
Annihilating polynomial 052M4SX57M
Eigenvalues are the roots of the characteristic polynomial 053QVHF6N7
A matrix is diagonalisable iff it has a basis of eigenvectors 056SW1E0AC
Spectral theorem 06J4PXRG22
Geometric multiplicity 06YSJVR356
Generalised eigenspace 07PNDWA7CE
Existence of an eigenvalue over an algebraically closed field 08CJB2NQ92
Minimal polynomial divides the characteristic polynomial 093X6MN6Y6
Eigenspace 0AYEPBQMAW
Eigenvector 0B5H4208Z9
Polynomial of an endomorphism 0BNEXSKNRV
Least squares 0BYTQ4X1HC
Characteristic polynomial of a linear map 0BZ81AJNWF
Determinant is the product of the eigenvalues 0C2F40MEBC
Eigenvalues of a self-adjoint operator are real 0C4NT4GT3X
Spectral theorem for Hermitian matrices 0DGZBQWZWN
Normal operator 0ENZN5G4CV
Invariant subspace 0EP59ESQ94
Gershgorin circle theorem 0ETGP0XVCJ
Cayley–Hamilton theorem 0F765KFC2J
Eigenvectors with distinct eigenvalues are linearly independent 0HNMFG8F8K
Eigenvalues of a positive definite matrix are positive 0HYC2RHD3M
Eigenspaces of a self-adjoint operator are orthogonal 0JHGAV9H25
Singular value 0K2TX12X58
Rayleigh quotient 0K4WCNTMHG
Determinant of a linear map is the determinant of its matrix 0KMN32XGPF
Nilpotent endomorphism 0M8MHYERTY
Self-adjoint operator 0MB8HQW495
Orthogonal projection is the closest point 0MCDDSCB9Z
Diagonalisable iff the minimal polynomial has distinct roots 0MPAJRB23T
Trace is the sum of the eigenvalues 0NMXDCNNKR
Restriction of an endomorphism to an invariant subspace 0PFPWKBDTJ
Principal axis theorem 0PR5P9Z4KM
Orthogonal projection 0Q8HZTJC7G
Cayley–Hamilton theorem for linear maps 0QEM68Y0DE
Eigenvalues are the roots of the minimal polynomial 0QF4ZEFG52
Semisimple endomorphism 0QS8P74F2Z
Projection 0RH4E3CAV9
Trace of a linear map 0RPDTWY2JS
Algebraic multiplicity 0S7A0EMPF4
Generalised eigenvector 0SAAHC36J9
Minimal polynomial 0V58FPM3EQ
Cartan–Dieudonné theorem 0VMWGYY35Q
Minimal polynomial divides every annihilating polynomial 0VTAD25A3G
Trace of a linear map is the trace of its matrix 0WCPH5Z7FA
Distinct eigenvalues imply diagonalisable 0Y95Z7MT02
Decomposition into generalised eigenspaces 0Z3GDY89T5
Eigenvalue 0ZND7NQ970

Sources

Web https://en.wikipedia.org/wiki/Endomorphism, lead section

What links here

Used by

Determinant of a linear map 03F5KP94K0
Polynomial of an endomorphism 0BNEXSKNRV
Characteristic polynomial of a linear map 0BZ81AJNWF
Normal operator 0ENZN5G4CV
Invariant subspace 0EP59ESQ94
Nilpotent endomorphism 0M8MHYERTY
Self-adjoint operator 0MB8HQW495
Projection 0RH4E3CAV9
Trace of a linear map 0RPDTWY2JS
Eigenvalue 0ZND7NQ970