SquareMatrix(ndim, R)ΒΆ

matrix.spad line 313 [edit on github]

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

#: % -> NonNegativeInteger

from Aggregate

*: (%, %) -> %

from LeftModule %

*: (%, DirectProduct(ndim, R)) -> DirectProduct(ndim, R)

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

*: (%, Integer) -> % if R has LinearlyExplicitOver Integer and R has Ring

from RightModule Integer

*: (%, R) -> %

from RightModule R

*: (DirectProduct(ndim, R), %) -> DirectProduct(ndim, R)

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

*: (Integer, %) -> % if % has AbelianGroup or R has AbelianGroup

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (R, %) -> %

from LeftModule R

+: (%, %) -> %

from AbelianSemiGroup

-: % -> % if % has AbelianGroup or R has AbelianGroup

from AbelianGroup

-: (%, %) -> % if % has AbelianGroup or 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))

any?: (R -> Boolean, %) -> Boolean

from HomogeneousAggregate 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 CoercibleFrom Fraction Integer

coerce: Integer -> % if R has Ring or R has RetractableTo Integer

from CoercibleFrom Integer

coerce: R -> %

from CoercibleFrom 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 -> Boolean, %) -> NonNegativeInteger

from HomogeneousAggregate R

count: (R, %) -> NonNegativeInteger

from HomogeneousAggregate R

D: % -> % if R has DifferentialRing and R has Ring

from DifferentialRing

D: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol and R has Ring

from PartialDifferentialRing Symbol

D: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol and R has Ring

from PartialDifferentialRing Symbol

D: (%, NonNegativeInteger) -> % if R has DifferentialRing and R has Ring

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 PartialDifferentialRing Symbol and R has Ring

from PartialDifferentialRing Symbol

D: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol and R has Ring

from PartialDifferentialRing Symbol

determinant: % -> R if R has CommutativeRing

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

diagonal?: % -> Boolean

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

diagonal: % -> DirectProduct(ndim, R)

from SquareMatrixCategory(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 DifferentialRing and R has Ring

from DifferentialRing

differentiate: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol and R has Ring

from PartialDifferentialRing Symbol

differentiate: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol and R has Ring

from PartialDifferentialRing Symbol

differentiate: (%, NonNegativeInteger) -> % if R has DifferentialRing and R has Ring

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 PartialDifferentialRing Symbol and R has Ring

from PartialDifferentialRing Symbol

differentiate: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol and R has Ring

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?: % -> Boolean

from Aggregate

empty: () -> %

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)

every?: (R -> Boolean, %) -> Boolean

from HomogeneousAggregate R

exquo: (%, R) -> Union(%, failed) if R has IntegralDomain

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

hash: % -> SingleInteger if R has Finite

from Hashable

hashUpdate!: (HashState, %) -> HashState if R has Finite

from Hashable

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))

max: % -> R if R has OrderedSet

from HomogeneousAggregate R

max: ((R, R) -> Boolean, %) -> R

from HomogeneousAggregate 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

members: % -> List R

from HomogeneousAggregate R

min: % -> R if R has OrderedSet

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

parts: % -> List R

from HomogeneousAggregate R

Pfaffian: % -> R if R has CommutativeRing

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

plenaryPower: (%, PositiveInteger) -> % if R has CommutativeRing

from NonAssociativeAlgebra 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 LinearlyExplicitOver Integer and R has Ring

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 LinearlyExplicitOver Integer and R has Ring

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) -> Boolean

from Aggregate

size: () -> NonNegativeInteger if R has Finite

from Finite

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 % has AbelianGroup or 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 % has AbelianGroup or R has AbelianGroup

CoercibleFrom Fraction Integer if R has RetractableTo Fraction Integer

CoercibleFrom Integer if R has RetractableTo Integer

CoercibleFrom R

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 DifferentialRing and R has Ring

Evalable R if R has Evalable R

Finite if R has Finite

finiteAggregate

FullyLinearlyExplicitOver R if R has Ring

FullyRetractableTo R

Hashable if R has Finite

HomogeneousAggregate R

InnerEvalable(R, R) if R has Evalable R

LeftModule %

LeftModule R

LinearlyExplicitOver Integer if R has LinearlyExplicitOver Integer and R has Ring

LinearlyExplicitOver R if R has Ring

Magma

MagmaWithUnit if R has SemiRing

Module R if R has CommutativeRing

Monoid if R has SemiRing

NonAssociativeAlgebra R if R has CommutativeRing

NonAssociativeRing if R has Ring

NonAssociativeRng if R has Ring

NonAssociativeSemiRing if R has SemiRing

NonAssociativeSemiRng

PartialDifferentialRing Symbol if R has PartialDifferentialRing Symbol and R has Ring

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 Integer if R has LinearlyExplicitOver Integer and R has Ring

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))

TwoSidedRecip

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