SquareMatrix(ndim, R)

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.

# : % -> NonNegativeInteger
from Aggregate
* : (%, %) -> %
from Magma
* : (%, R) -> %
from RightModule(R)
* : (%, Integer) -> % if R has LinearlyExplicitOver(Integer) and R has Ring
from RightModule(Integer)
* : (R, %) -> %
from LeftModule(R)
* : (Integer, %) -> % if R has AbelianGroup or % has AbelianGroup
from AbelianGroup
* : (NonNegativeInteger, %) -> %
from AbelianMonoid
* : (PositiveInteger, %) -> %
from AbelianSemiGroup
* : (%, DirectProduct(ndim, R)) -> DirectProduct(ndim, R)
from SquareMatrixCategory(ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
* : (DirectProduct(ndim, R), %) -> DirectProduct(ndim, R)
from SquareMatrixCategory(ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
+ : (%, %) -> %
from AbelianSemiGroup
- : % -> % if R has AbelianGroup or % has AbelianGroup
from AbelianGroup
- : (%, %) -> % if R has AbelianGroup or % has AbelianGroup
from AbelianGroup
/ : (%, R) -> % if R has Field
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
0 : () -> %
from AbelianMonoid
1 : () -> % if R has SemiRing
from MagmaWithUnit
= : (%, %) -> Boolean
from BasicType
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 : (%, Mapping(R, R)) -> % if R has Ring
from DifferentialExtension(R)
D : (%, Mapping(R, R), NonNegativeInteger) -> % if R has Ring
from DifferentialExtension(R)
D : (%, NonNegativeInteger) -> % if R has DifferentialRing and R has Ring
from DifferentialRing
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)
Pfaffian : % -> R if R has CommutativeRing
from SquareMatrixCategory(ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
^ : (%, 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
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? : (Mapping(Boolean, R), %) -> Boolean
from HomogeneousAggregate(R)
associator : (%, %, %) -> % if R has Ring
from NonAssociativeRng
characteristic : () -> NonNegativeInteger if R has Ring
from NonAssociativeRing
coerce : R -> %
from Algebra(R)
coerce : Fraction(Integer) -> % if R has RetractableTo(Fraction(Integer))
from CoercibleFrom(Fraction(Integer))
coerce : Integer -> % if R has RetractableTo(Integer) or 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)
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)
count : (Mapping(Boolean, R), %) -> NonNegativeInteger
from HomogeneousAggregate(R)
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 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 : (%, Mapping(R, R)) -> % if R has Ring
from DifferentialExtension(R)
differentiate : (%, Mapping(R, R), NonNegativeInteger) -> % if R has Ring
from DifferentialExtension(R)
differentiate : (%, NonNegativeInteger) -> % if R has DifferentialRing and R has Ring
from DifferentialRing
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 : () -> %
from Aggregate
empty? : % -> Boolean
from Aggregate
enumerate : () -> List(%) if R has Finite
from Finite
eq? : (%, %) -> Boolean
from Aggregate
eval : (%, R, R) -> % if R has Evalable(R)
from InnerEvalable(R, R)
eval : (%, 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 : (%, List(Equation(R))) -> % if R has Evalable(R)
from Evalable(R)
every? : (Mapping(Boolean, R), %) -> 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 : (Mapping(R, R), %) -> %
from RectangularMatrixCategory(ndim, ndim, R, DirectProduct(ndim, R), DirectProduct(ndim, R))
map : (Mapping(R, 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 : (Mapping(Boolean, R, R), %) -> 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))
minRowIndex : % -> 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))
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))
nullSpace : % -> List(DirectProduct(ndim, R)) if R has IntegralDomain
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))
one? : % -> Boolean if R has SemiRing
from MagmaWithUnit
opposite? : (%, %) -> Boolean
from AbelianMonoid
parts : % -> List(R)
from HomogeneousAggregate(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(%) -> Matrix(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(%), Vector(%)) -> Record(mat : Matrix(R), vec : Vector(R)) if R has Ring
from LinearlyExplicitOver(R)
reducedSystem : (Matrix(%), Vector(%)) -> Record(mat : Matrix(Integer), vec : Vector(Integer)) if R has LinearlyExplicitOver(Integer) and R has Ring
from LinearlyExplicitOver(Integer)
retract : % -> R
from RetractableTo(R)
retract : % -> Fraction(Integer) if R has RetractableTo(Fraction(Integer))
from RetractableTo(Fraction(Integer))
retract : % -> Integer if R has RetractableTo(Integer)
from RetractableTo(Integer)
retractIfCan : % -> Union(R, "failed")
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)
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 or % 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
~= : (%, %) -> Boolean
from BasicType

NonAssociativeSemiRing

LeftModule(R)

Evalable(R)

ConvertibleTo(InputForm)

Rng

CoercibleFrom(Integer)

TwoSidedRecip

FullyRetractableTo(R)

SemiRing

unitsKnown

FullyLinearlyExplicitOver(R)

RetractableTo(Fraction(Integer))

Magma

SemiGroup

LeftModule(%)

NonAssociativeRing

finiteAggregate

PartialDifferentialRing(Symbol)

Module(R)

BiModule(%, %)

DifferentialRing

BiModule(R, R)

Algebra(R)

RightModule(R)

CoercibleTo(Matrix(R))

InnerEvalable(R, R)

NonAssociativeSemiRng

CancellationAbelianMonoid

RetractableTo(Integer)

SetCategory

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

AbelianMonoid

MagmaWithUnit

Comparable

RightModule(%)

Hashable

HomogeneousAggregate(R)

CoercibleTo(OutputForm)

LinearlyExplicitOver(Integer)

DifferentialExtension(R)

Ring

SemiRng

Monoid

Finite

Aggregate

BasicType

RightModule(Integer)

AbelianSemiGroup

CoercibleFrom(Fraction(Integer))

LinearlyExplicitOver(R)

NonAssociativeRng

CoercibleFrom(R)

RetractableTo(R)

AbelianGroup

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

NonAssociativeAlgebra(R)