FiniteRankAlgebra(R, UP)ΒΆ

algcat.spad line 64

A FiniteRankAlgebra is an algebra over a commutative ring R which is a free R-module of finite rank.

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
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
characteristicPolynomial: % -> UP
characteristicPolynomial(a) returns the characteristic polynomial of the regular representation of a with respect to any basis.
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
coordinates: (%, Vector %) -> Vector R
coordinates(a, basis) returns the coordinates of a with respect to the basis basis.
coordinates: (Vector %, Vector %) -> Matrix R
coordinates([v1, ..., vm], basis) returns the coordinates of the vi's with to the basis basis. The coordinates of vi are contained in the ith row of the matrix returned by this function.
discriminant: Vector % -> R
discriminant([v1, .., vn]) returns determinant(traceMatrix([v1, .., vn])).
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
latex: % -> String
from SetCategory
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
minimalPolynomial: % -> UP if R has Field
minimalPolynomial(a) returns the minimal polynomial of a.
norm: % -> R
norm(a) returns the determinant of the regular representation of a with respect to any basis.
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
rank: () -> PositiveInteger
rank() returns the rank of the algebra.
recip: % -> Union(%, failed)
from MagmaWithUnit
regularRepresentation: (%, Vector %) -> Matrix R
regularRepresentation(a, basis) returns the matrix m of the linear map defined by left multiplication by a with respect to the basis basis. That is for all x we have coordinates(a*x, basis) = m*coordinates(x, basis).
represents: (Vector R, Vector %) -> %
represents([a1, .., an], [v1, .., vn]) returns a1*v1 + ... + an*vn.
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
sample: %
from AbelianMonoid
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
trace: % -> R
trace(a) returns the trace of the regular representation of a with respect to any basis.
traceMatrix: Vector % -> Matrix R
traceMatrix([v1, .., vn]) is the n-by-n matrix ( Tr(vi * vj) )
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra R

BasicType

BiModule(%, %)

BiModule(R, R)

CancellationAbelianMonoid

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleTo OutputForm

LeftModule %

LeftModule R

Magma

MagmaWithUnit

Module R

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

RightModule %

RightModule R

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown