Linear and multilinear algebra; matrix theory Definition
Endomorphism
Builds on Linear map
Definition
Let be a field and let be a vector space over . An endomorphism of is a linear map . The set of all endomorphisms of is written .
Remarks
An endomorphism is also called a linear operator on . Endomorphisms can be composed with each other, so their powers and polynomials are defined. Under composition and addition, is the algebra of endomorphisms. Eigenvalues, the trace and the determinant are defined for endomorphisms of finite-dimensional spaces.
Reading path
To understand this page from the ground up, read these 3 pages in order. The foundations come first.
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- Vector space definition
- Linear map definition
- Endomorphism definition
Where this leads
117 pages build on this one.
- Singular value decomposition
- Jordan–Chevalley decomposition
- Reflection
- Determinant of a linear map
- Spectrum of a matrix
- Annihilating polynomial
- Eigenvalues are the roots of the characteristic polynomial
- A matrix is diagonalisable iff it has a basis of eigenvectors
- Spectral theorem
- Geometric multiplicity
- Generalised eigenspace
- Existence of an eigenvalue over an algebraically closed field
- Minimal polynomial divides the characteristic polynomial
- Eigenspace
- Eigenvector
- Polynomial of an endomorphism
- Least squares
- Characteristic polynomial of a linear map
- Determinant is the product of the eigenvalues
- Eigenvalues of a self-adjoint operator are real
- Spectral theorem for Hermitian matrices
- Normal operator
- Invariant subspace
- Gershgorin circle theorem
- Cayley–Hamilton theorem
- Eigenvectors with distinct eigenvalues are linearly independent
- Eigenvalues of a positive definite matrix are positive
- Eigenspaces of a self-adjoint operator are orthogonal
- Singular value
- Rayleigh quotient
- Determinant of a linear map is the determinant of its matrix
- Nilpotent endomorphism
- Self-adjoint operator
- Orthogonal projection is the closest point
- Diagonalisable iff the minimal polynomial has distinct roots
- Trace is the sum of the eigenvalues
- Restriction of an endomorphism to an invariant subspace
- Principal axis theorem
- Orthogonal projection
- Cayley–Hamilton theorem for linear maps
- Eigenvalues are the roots of the minimal polynomial
- Semisimple endomorphism
- Projection
- Trace of a linear map
- Algebraic multiplicity
- Generalised eigenvector
- Minimal polynomial
- Cartan–Dieudonné theorem
- Minimal polynomial divides every annihilating polynomial
- Trace of a linear map is the trace of its matrix
- Distinct eigenvalues imply diagonalisable
- Decomposition into generalised eigenspaces
- Eigenvalue
- Non-negative least squares
- Conjugate residual method
- Local regression
- Beurling's theorem
- Hilbert–Schmidt expansion theorem
- Minimal residual method
- Toeplitz operator
- Céa's lemma
- Bilinear series of a symmetric kernel
- Ordinary least squares
- Field trace
- Smoothing spline
- Weil conjectures
- Lax–Milgram theorem
- Central polynomial
- Field norm
- Kaplansky's theorem on PI rings
- Principal curvature
- Shape operator
- Kalman rank condition
- Response surface methodology
- Nonlinear regression
- Spectral theorem for compact self-adjoint operators
- Courant–Fischer characterization of characteristic values
- Generalized minimal residual method
- Minimum total potential energy principle
- Arnold's cat map
- Gauss–Newton algorithm
- Standard conjectures on algebraic cycles
- Hilbert space adjoint
- Trace form
- Lidskii's trace formula for integral operators
- Posner's theorem
- Spectrum of the Laplace–Beltrami operator
- Hilbert projection theorem
- Eckart–Young theorem
- Ridge regression
- Zermelo's navigation problem
- L-curve
- Biconjugate gradient stabilized method
- Galerkin method
- Operator monotone function
- Empirical risk minimization
- Grothendieck–Lefschetz trace formula
- Isotonic regression
- Projection body
- Mercer's theorem
- Principal component analysis
- Bidiagonalization
- Ladyzhenskaya–Babuška–Brezzi condition
- Discriminant of a field extension
- Levenberg–Marquardt algorithm
- Schur product theorem
- Riesz representation theorem
- Box–Jenkins method
- Singular values of a compact operator
- Lasso
- Householder transformation
- Numerical methods for linear least squares
- Schmidt's singular value decomposition of a kernel
- Pseudo-Anosov map
- Amitsur–Levitzki theorem
- Hilbert's Theorem 90
- Time-optimal control
Further reading
- Web https://en.wikipedia.org/wiki/Endomorphism, lead section
About this page
| Tag | 0PTF1PTJK0 |
|---|---|
| URI | m2://0PTF1PTJK0 |
| Kind | definition |
| Writing | W0 stub |
| MSC | 15A04 |
| Defines | Endomorphism |
| Uses | Linear map |
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