SquareMatrix(ndim, R)ΒΆ

matrix.spad line 319

SquareMatrix is a matrix domain of square matrices, where the number of rows (= number of columns) is a parameter of the type.

0: %
from AbelianMonoid
1: % if R has SemiRing
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, DirectProduct(ndim, R)) -> DirectProduct(ndim, R)
from SquareMatrixCategory(ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
*: (%, R) -> %
from RightModule R
*: (DirectProduct(ndim, R), %) -> DirectProduct(ndim, R)
from SquareMatrixCategory(ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
*: (Integer, %) -> % if R has AbelianGroup
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (R, %) -> %
from LeftModule R
+: (%, %) -> %
from AbelianSemiGroup
-: % -> % if R has AbelianGroup
from AbelianGroup
-: (%, %) -> % if R has AbelianGroup
from AbelianGroup
/: (%, R) -> % if R has Field
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
=: (%, %) -> Boolean
from BasicType
^: (%, Integer) -> % if R has Field
from SquareMatrixCategory(ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
^: (%, NonNegativeInteger) -> % if R has SemiRing
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
annihilate?: (%, %) -> Boolean if R has Ring
from Rng
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
antisymmetric?: % -> Boolean
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
associator: (%, %, %) -> % if R has Ring
from NonAssociativeRng
characteristic: () -> NonNegativeInteger if R has Ring
from NonAssociativeRing
coerce: % -> Matrix R
coerce(m) converts a matrix of type SquareMatrix to a matrix of type Matrix.
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> % if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer
coerce: Integer -> % if R has RetractableTo Integer or R has Ring
from NonAssociativeRing
coerce: R -> %
from Algebra R
column: (%, Integer) -> DirectProduct(ndim, R)
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
columnSpace: % -> List DirectProduct(ndim, R) if R has EuclideanDomain
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
commutator: (%, %) -> % if R has Ring
from NonAssociativeRng
convert: % -> InputForm if R has ConvertibleTo InputForm
from ConvertibleTo InputForm
copy: % -> %
from Aggregate
count: (R, %) -> NonNegativeInteger
from HomogeneousAggregate R
D: % -> % if R has Ring and R has DifferentialRing
from DifferentialRing
D: (%, List Symbol) -> % if R has Ring and R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
D: (%, List Symbol, List NonNegativeInteger) -> % if R has Ring and R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
D: (%, NonNegativeInteger) -> % if R has Ring and R has DifferentialRing
from DifferentialRing
D: (%, R -> R) -> % if R has Ring
from DifferentialExtension R
D: (%, R -> R, NonNegativeInteger) -> % if R has Ring
from DifferentialExtension R
D: (%, Symbol) -> % if R has Ring and R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
D: (%, Symbol, NonNegativeInteger) -> % if R has Ring and R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
determinant: % -> R if R has CommutativeRing
from SquareMatrixCategory(ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
diagonal: % -> DirectProduct(ndim, R)
from SquareMatrixCategory(ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
diagonal?: % -> Boolean
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
diagonalMatrix: List R -> %
from SquareMatrixCategory(ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
diagonalProduct: % -> R
from SquareMatrixCategory(ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
differentiate: % -> % if R has Ring and R has DifferentialRing
from DifferentialRing
differentiate: (%, List Symbol) -> % if R has Ring and R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: (%, List Symbol, List NonNegativeInteger) -> % if R has Ring and R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: (%, NonNegativeInteger) -> % if R has Ring and R has DifferentialRing
from DifferentialRing
differentiate: (%, R -> R) -> % if R has Ring
from DifferentialExtension R
differentiate: (%, R -> R, NonNegativeInteger) -> % if R has Ring
from DifferentialExtension R
differentiate: (%, Symbol) -> % if R has Ring and R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: (%, Symbol, NonNegativeInteger) -> % if R has Ring and R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
elt: (%, Integer, Integer) -> R
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
elt: (%, Integer, Integer, R) -> R
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
empty: () -> %
from Aggregate
empty?: % -> Boolean
from Aggregate
enumerate: () -> List % if R has Finite
from Finite
eq?: (%, %) -> Boolean
from Aggregate
eval: (%, Equation R) -> % if R has Evalable R
from Evalable R
eval: (%, List Equation R) -> % if R has Evalable R
from Evalable R
eval: (%, List R, List R) -> % if R has Evalable R
from InnerEvalable(R, R)
eval: (%, R, R) -> % if R has Evalable R
from InnerEvalable(R, R)
exquo: (%, R) -> Union(%, failed) if R has IntegralDomain
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
index: PositiveInteger -> % if R has Finite
from Finite
inverse: % -> Union(%, failed) if R has Field
from SquareMatrixCategory(ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
latex: % -> String
from SetCategory
leftPower: (%, NonNegativeInteger) -> % if R has SemiRing
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed) if R has SemiRing
from MagmaWithUnit
less?: (%, NonNegativeInteger) -> Boolean
from Aggregate
listOfLists: % -> List List R
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
lookup: % -> PositiveInteger if R has Finite
from Finite
map: ((R, R) -> R, %, %) -> %
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
map: (R -> R, %) -> %
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
matrix: List List R -> %
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
maxColIndex: % -> Integer
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
maxRowIndex: % -> Integer
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
member?: (R, %) -> Boolean
from HomogeneousAggregate R
minColIndex: % -> Integer
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
minordet: % -> R if R has CommutativeRing
from SquareMatrixCategory(ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
minRowIndex: % -> Integer
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
more?: (%, NonNegativeInteger) -> Boolean
from Aggregate
ncols: % -> NonNegativeInteger
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
nrows: % -> NonNegativeInteger
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
nullity: % -> NonNegativeInteger if R has IntegralDomain
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
nullSpace: % -> List DirectProduct(ndim, R) if R has IntegralDomain
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
one?: % -> Boolean if R has SemiRing
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
Pfaffian: % -> R if R has CommutativeRing
from SquareMatrixCategory(ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
qelt: (%, Integer, Integer) -> R
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
random: () -> % if R has Finite
from Finite
rank: % -> NonNegativeInteger if R has IntegralDomain
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
recip: % -> Union(%, failed) if R has SemiRing
from MagmaWithUnit
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if R has Ring and R has LinearlyExplicitOver Integer
from LinearlyExplicitOver Integer
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix R, vec: Vector R) if R has Ring
from LinearlyExplicitOver R
reducedSystem: Matrix % -> Matrix Integer if R has Ring and R has LinearlyExplicitOver Integer
from LinearlyExplicitOver Integer
reducedSystem: Matrix % -> Matrix R if R has Ring
from LinearlyExplicitOver R
retract: % -> Fraction Integer if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer
retract: % -> Integer if R has RetractableTo Integer
from RetractableTo Integer
retract: % -> R
from RetractableTo R
retractIfCan: % -> Union(Fraction Integer, failed) if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed) if R has RetractableTo Integer
from RetractableTo Integer
retractIfCan: % -> Union(R, failed)
from RetractableTo R
rightPower: (%, NonNegativeInteger) -> % if R has SemiRing
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed) if R has SemiRing
from MagmaWithUnit
row: (%, Integer) -> DirectProduct(ndim, R)
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
rowEchelon: % -> % if R has EuclideanDomain
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
sample: %
from AbelianMonoid
scalarMatrix: R -> %
from SquareMatrixCategory(ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
size: () -> NonNegativeInteger if R has Finite
from Finite
size?: (%, NonNegativeInteger) -> Boolean
from Aggregate
smaller?: (%, %) -> Boolean if R has Finite
from Comparable
square?: % -> Boolean
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
squareMatrix: Matrix R -> %
squareMatrix(m) converts a matrix of type Matrix to a matrix of type SquareMatrix.
subtractIfCan: (%, %) -> Union(%, failed) if R has AbelianGroup
from CancellationAbelianMonoid
symmetric?: % -> Boolean
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
trace: % -> R
from SquareMatrixCategory(ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
transpose: % -> %
transpose(m) returns the transpose of the matrix m.
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup if R has AbelianGroup

AbelianMonoid

AbelianSemiGroup

Aggregate

Algebra R if R has CommutativeRing

BasicType

BiModule(%, %)

BiModule(R, R)

CancellationAbelianMonoid if R has AbelianGroup

CoercibleTo Matrix R

CoercibleTo OutputForm

Comparable if R has Finite

ConvertibleTo InputForm if R has ConvertibleTo InputForm

DifferentialExtension R if R has Ring

DifferentialRing if R has Ring and R has DifferentialRing

Evalable R if R has Evalable R

Finite if R has Finite

finiteAggregate

FullyLinearlyExplicitOver R if R has Ring

FullyRetractableTo R

HomogeneousAggregate R

InnerEvalable(R, R) if R has Evalable R

LeftModule %

LeftModule R

LinearlyExplicitOver Integer if R has Ring and R has LinearlyExplicitOver Integer

LinearlyExplicitOver R if R has Ring

Magma

MagmaWithUnit if R has SemiRing

Module R if R has CommutativeRing

Monoid if R has SemiRing

NonAssociativeRing if R has Ring

NonAssociativeRng if R has Ring

NonAssociativeSemiRing if R has SemiRing

NonAssociativeSemiRng

PartialDifferentialRing Symbol if R has Ring and R has PartialDifferentialRing Symbol

RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer

RetractableTo R

RightModule %

RightModule R

Ring if R has Ring

Rng if R has Ring

SemiGroup

SemiRing if R has SemiRing

SemiRng

SetCategory

SquareMatrixCategory(ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))

unitsKnown if R has Ring or R has CommutativeStar and R has unitsKnown