# FramedAlgebra(R, UP)¶

A FramedAlgebra is a FiniteRankAlgebra together with a fixed `R`

-module basis.

- 0: %
- from AbelianMonoid
- 1: %
- from MagmaWithUnit
- *: (%, %) -> %
- from Magma
- *: (%, R) -> %
- from RightModule R
- *: (Integer, %) -> %
- from AbelianGroup
- *: (NonNegativeInteger, %) -> %
- from AbelianMonoid
- *: (PositiveInteger, %) -> %
- from AbelianSemiGroup
- *: (R, %) -> %
- from LeftModule R
- +: (%, %) -> %
- from AbelianSemiGroup
- -: % -> %
- from AbelianGroup
- -: (%, %) -> %
- from AbelianGroup
- =: (%, %) -> Boolean
- from BasicType
- ^: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
- from Magma
- ~=: (%, %) -> Boolean
- from BasicType
- annihilate?: (%, %) -> Boolean
- from Rng
- antiCommutator: (%, %) -> %
- from NonAssociativeSemiRng
- associator: (%, %, %) -> %
- from NonAssociativeRng
- basis: () -> Vector %
- from FramedModule R
- characteristic: () -> NonNegativeInteger
- from NonAssociativeRing
- characteristicPolynomial: % -> UP
- from FiniteRankAlgebra(R, UP)
- charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero
- from CharacteristicNonZero
- coerce: % -> OutputForm
- from CoercibleTo OutputForm
- coerce: Integer -> %
- from NonAssociativeRing
- coerce: R -> %
- from Algebra R
- commutator: (%, %) -> %
- from NonAssociativeRng
- convert: % -> InputForm if R has Finite
- from ConvertibleTo InputForm
- convert: % -> Vector R
- from FramedModule R
- convert: Vector R -> %
- from FramedModule R
- coordinates: % -> Vector R
- from FramedModule R
- coordinates: (%, Vector %) -> Vector R
- from FiniteRankAlgebra(R, UP)
- coordinates: (Vector %, Vector %) -> Matrix R
- from FiniteRankAlgebra(R, UP)
- coordinates: Vector % -> Matrix R
- from FramedModule R

- discriminant: () -> R
`discriminant()`

= determinant(traceMatrix()).- discriminant: Vector % -> R
- from FiniteRankAlgebra(R, UP)
- enumerate: () -> List % if R has Finite
- from Finite
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory
- index: PositiveInteger -> % if R has Finite
- from Finite
- latex: % -> String
- from SetCategory
- leftPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- leftPower: (%, PositiveInteger) -> %
- from Magma
- leftRecip: % -> Union(%, failed)
- from MagmaWithUnit
- lookup: % -> PositiveInteger if R has Finite
- from Finite
- minimalPolynomial: % -> UP if R has Field
- from FiniteRankAlgebra(R, UP)
- norm: % -> R
- from FiniteRankAlgebra(R, UP)
- one?: % -> Boolean
- from MagmaWithUnit
- opposite?: (%, %) -> Boolean
- from AbelianMonoid
- random: () -> % if R has Finite
- from Finite
- rank: () -> PositiveInteger
- from FramedModule R
- recip: % -> Union(%, failed)
- from MagmaWithUnit

- regularRepresentation: % -> Matrix R
`regularRepresentation(a)`

returns the matrix`m`

of the linear map defined by left multiplication by`a`

with respect to the fixed basis. That is for all`x`

we have`coordinates(a*x) = m*coordinates(x)`

.- regularRepresentation: (%, Vector %) -> Matrix R
- from FiniteRankAlgebra(R, UP)
- represents: (Vector R, Vector %) -> %
- from FiniteRankAlgebra(R, UP)
- represents: Vector R -> %
- from FramedModule R
- rightPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- rightPower: (%, PositiveInteger) -> %
- from Magma
- rightRecip: % -> Union(%, failed)
- from MagmaWithUnit
- sample: %
- from AbelianMonoid
- size: () -> NonNegativeInteger if R has Finite
- from Finite
- smaller?: (%, %) -> Boolean if R has Finite
- from Comparable
- subtractIfCan: (%, %) -> Union(%, failed)
- from CancellationAbelianMonoid
- trace: % -> R
- from FiniteRankAlgebra(R, UP)

- traceMatrix: () -> Matrix R
`traceMatrix()`

is the`n`

-by-`n`

matrix (`Tr(vi * vj)`

), where`v1`

, …,`vn`

are the elements of the fixed basis.- traceMatrix: Vector % -> Matrix R
- from FiniteRankAlgebra(R, UP)
- zero?: % -> Boolean
- from AbelianMonoid

Algebra R

BiModule(%, %)

BiModule(R, R)

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

Comparable if R has Finite

ConvertibleTo InputForm if R has Finite

FiniteRankAlgebra(R, UP)

Module R