SquareMatrixCategory(ndim, R, Row, Col)ΒΆ

matcat.spad line 908

SquareMatrixCategory is a general square matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col.

0: %
from AbelianMonoid
1: % if R has SemiRing
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, Col) -> Col
x * c is the product of the matrix x and the column vector c. Error: if the dimensions are incompatible.
*: (%, R) -> %
from RightModule R
*: (Integer, %) -> % if R has AbelianGroup
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (R, %) -> %
from LeftModule R
*: (Row, %) -> Row
r * x is the product of the row vector r and the matrix x. Error: if the dimensions are incompatible.
+: (%, %) -> %
from AbelianSemiGroup
-: % -> % if R has AbelianGroup
from AbelianGroup
-: (%, %) -> % if R has AbelianGroup
from AbelianGroup
/: (%, R) -> % if R has Field
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
=: (%, %) -> Boolean
from BasicType
^: (%, Integer) -> % if R has Field
m^n computes an integral power of the matrix m. Error: if the matrix is not invertible.
^: (%, 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, Row, Col)
associator: (%, %, %) -> % if R has Ring
from NonAssociativeRng
characteristic: () -> NonNegativeInteger if R has Ring
from NonAssociativeRing
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> % if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer
coerce: Integer -> % if R has Ring or R has RetractableTo Integer
from NonAssociativeRing
coerce: R -> %
from Algebra R
column: (%, Integer) -> Col
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
columnSpace: % -> List Col if R has EuclideanDomain
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
commutator: (%, %) -> % if R has Ring
from NonAssociativeRng
convert: % -> InputForm if R has Finite
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
determinant(m) returns the determinant of the matrix m.
diagonal: % -> Row
diagonal(m) returns a row consisting of the elements on the diagonal of the matrix m.
diagonal?: % -> Boolean
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
diagonalMatrix: List R -> %
diagonalMatrix(l) returns a diagonal matrix with the elements of l on the diagonal.
diagonalProduct: % -> R
diagonalProduct(m) returns the product of the elements on the diagonal of the matrix m.
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, Row, Col)
elt: (%, Integer, Integer, R) -> R
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
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, Row, Col)
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
index: PositiveInteger -> % if R has Finite
from Finite
inverse: % -> Union(%, failed) if R has Field
inverse(m) returns the inverse of the matrix m, if that matrix is invertible and returns “failed” otherwise.
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, Row, Col)
lookup: % -> PositiveInteger if R has Finite
from Finite
map: ((R, R) -> R, %, %) -> %
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
map: (R -> R, %) -> %
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
matrix: List List R -> %
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
maxColIndex: % -> Integer
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
maxRowIndex: % -> Integer
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
member?: (R, %) -> Boolean
from HomogeneousAggregate R
minColIndex: % -> Integer
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
minordet: % -> R if R has CommutativeRing
minordet(m) computes the determinant of the matrix m using minors.
minRowIndex: % -> Integer
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
more?: (%, NonNegativeInteger) -> Boolean
from Aggregate
ncols: % -> NonNegativeInteger
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
nrows: % -> NonNegativeInteger
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
nullity: % -> NonNegativeInteger if R has IntegralDomain
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
nullSpace: % -> List Col if R has IntegralDomain
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
one?: % -> Boolean if R has SemiRing
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
Pfaffian: % -> R if R has CommutativeRing
Pfaffian(m) returns the Pfaffian of the matrix m. Error: if the matrix is not antisymmetric.
qelt: (%, Integer, Integer) -> R
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
random: () -> % if R has Finite
from Finite
rank: % -> NonNegativeInteger if R has IntegralDomain
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
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) -> Row
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
rowEchelon: % -> % if R has EuclideanDomain
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
sample: %
from AbelianMonoid
scalarMatrix: R -> %
scalarMatrix(r) returns an n-by-n matrix with r's on the diagonal and zeroes elsewhere.
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, Row, Col)
subtractIfCan: (%, %) -> Union(%, failed) if R has AbelianGroup
from CancellationAbelianMonoid
symmetric?: % -> Boolean
from RectangularMatrixCategory(ndim, ndim, R, Row, Col)
trace: % -> R
trace(m) returns the trace of the matrix m. this is the sum of the elements on the diagonal 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 OutputForm

Comparable if R has Finite

ConvertibleTo InputForm if R has Finite

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, Row, Col)

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

unitsKnown if R has Ring