Linear and multilinear algebra; matrix theory Definition
Alternating multilinear map
Builds on Multilinear map
Definition
Let be a field, let , and let and be vector spaces over . A multilinear map is alternating when for each family with for some .
Remarks
An alternating map changes sign when two arguments are exchanged. If the characteristic of is not , then this sign rule implies that the map is alternating. An alternating map is zero on each linearly dependent family. The determinant is an alternating multilinear map of the rows of a square matrix.
Reading path
To understand this page from the ground up, read these 4 pages in order. The foundations come first.
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- Vector space definition
- Linear map definition
- Multilinear map definition
- Alternating multilinear map definition
Where this leads
157 pages build on this one.
- Scaling a row scales the determinant
- Leibniz formula for determinants
- Determinant of a scalar multiple
- Adding a multiple of one row to another preserves the determinant
- Triangularisation
- Characteristic polynomial
- Determinant of a linear map
- A matrix is invertible iff its determinant is a unit
- Rational canonical form
- Determinant is multiplicative
- Determinant as signed volume
- Product of a matrix and its adjugate
- Eigenvalues are the roots of the characteristic polynomial
- Laplace expansion along a column
- Determinant of the inverse
- Existence of an eigenvalue over an algebraically closed field
- Cauchy–Binet formula
- Minimal polynomial divides the characteristic polynomial
- Cramer's rule
- Permuting rows multiplies the determinant by the sign
- Determinant of a matrix exponential
- Rotation matrix
- Matrix determinant lemma
- Characteristic polynomial of a linear map
- Laplace expansion along a row
- Determinant is the product of the eigenvalues
- Determinant
- Special orthogonal group
- Inverse in terms of the adjugate
- Cayley–Hamilton theorem
- Trace is a coefficient of the characteristic polynomial
- Determinant of a matrix with a zero row
- Sylvester's determinant theorem
- Adjugate matrix
- Rule of Sarrus
- Determinant of a block triangular matrix
- Determinant of a triangular matrix
- Determinant of a matrix with two equal rows
- Determinant via the Schur complement
- Determinant of a linear map is the determinant of its matrix
- Trace is the sum of the eigenvalues
- Vandermonde determinant
- Determinant is multilinear in the rows
- Orientation of a real vector space
- Cayley–Hamilton theorem for linear maps
- Algebraic multiplicity
- Minor
- Determinant is a coefficient of the characteristic polynomial
- Characteristic polynomial does not depend on the basis
- Determinant of a family of vectors in a basis
- Special linear group
- Companion matrix
- Cofactor
- Sylvester's criterion
- Determinant of the identity matrix
- Determinant of a 2 by 2 matrix
- Determinant of a transpose
- Rotation group SO(3)
- Horocycle flow
- List of simple Lie groups
- Wigner D-matrix
- Jacobi's formula
- Oscillator representation
- Eisenstein series
- Schwarz–Christoffel mapping
- Maass wave form
- Modular lambda function
- The special linear group is the kernel of the determinant
- Modular curve
- Fuchsian group
- Representation theory of SL2(R)
- Compact symplectic group
- Siegel upper half-space
- Geodesic flow on a locally symmetric space
- Theta correspondence
- 6-j symbol
- Table of Lie groups
- Theta function
- Eisenstein series as a double series
- Sylvester's determinant identity
- Ramanujan tau function
- Siegel modular form
- Gaussian curvature
- Metaplectic group
- Resultant
- Central polynomial
- Field norm
- Kaplansky's theorem on PI rings
- Cayley–Menger determinant
- Poincaré group
- Hausdorff paradox
- Kalman rank condition
- Representation theory of SU(2)
- 3-j symbol
- Selberg's 1/4 conjecture
- Arthur–Selberg trace formula
- Selberg trace formula
- Linear canonical transformation
- Arithmetic group
- Baily–Borel compactification
- Schur polynomial
- Margulis superrigidity theorem
- Wigner's classification
- SU(2)
- Congruence subgroup
- Banach–Tarski paradox
- ADE singularity
- Wishart distribution
- Posner's theorem
- Inoue surface
- Weyl–Titchmarsh theory
- SL(2,C)
- Spin group
- Ultraparallel theorem
- Zimmer's conjecture
- j-invariant
- Zermelo's navigation problem
- Dedekind eta function
- Image of J
- Modular form
- Fisher's inequality
- Kleinian group
- Exceptional Lie group E8
- Petersson inner product
- Jeffreys prior
- Nielsen–Thurston classification
- Horocycle
- Classical group
- Binary quadratic form
- SL(2,R)
- Lorentz group
- Wynn's epsilon algorithm
- Representation theory of the Lorentz group
- Poincaré half-plane model
- Shanks transformation
- Cusp form
- Fundamental domain of the modular group
- Topological modular forms
- Discriminant of a field extension
- Nome
- Pin group
- Masur–Veech theorem
- Modular group
- Rogers–Ramanujan continued fraction
- Upper half-plane
- Hecke operator
- J-homomorphism
- Theta transformation formula
- Schramm–Loewner evolution
- Box spline
- de Branges space
- Amitsur–Levitzki theorem
- Hilbert's Theorem 90
- Nevanlinna parametrization of solutions
- Time-optimal control
- Loewner differential equation
- Exceptional Lie group G2
Further reading
About this page
| Tag | 0MSWYWHYF5 |
|---|---|
| URI | m2://0MSWYWHYF5 |
| Kind | definition |
| Writing | W0 stub |
| MSC | 15A69, 15A75 |
| Defines | Alternating multilinear map |
| Uses | Multilinear map |
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