nzmath.matrix.RingSquareMatrix Class Reference

Inheritance diagram for nzmath.matrix.RingSquareMatrix:

Collaboration diagram for nzmath.matrix.RingSquareMatrix:

Public Member Functions
def	__init__ (self, row, column=0, compo=0, coeff_ring=0)
def	__pow__ (self, other)
def	toFieldMatrix (self)
def	getRing (self)
def	isOrthogonalMatrix (self)
def	isAlternatingMatrix (self)
def	isSingular (self)
def	trace (self)
def	determinant (self)
def	cofactor (self, i, j)
def	commutator (self, other)
def	characteristicMatrix (self)
def	characteristicPolynomial (self)
def	adjugateMatrix (self)
def	cofactorMatrix (self)
def	smithNormalForm (self)
def	extsmithNormalForm (self)
Public Member Functions inherited from nzmath.matrix.SquareMatrix
def	isUpperTriangularMatrix (self)
def	isLowerTriangularMatrix (self)
def	isDiagonalMatrix (self)
def	isScalarMatrix (self)
def	isSymmetricMatrix (self)
Public Member Functions inherited from nzmath.matrix.Matrix
def	__getitem__ (self, index)
def	__setitem__ (self, key, value)
def	__eq__ (self, other)
def	__hash__ (self)
def	__ne__ (self, other)
def	__nonzero__ (self)
def	__contains__ (self, item)
def	__repr__ (self)
def	__str__ (self)
def	__call__ (self, arg)
def	map (self, function)
def	reduce (self, function, initializer=None)
def	copy (self)
def	set (self, compo)
def	setRow (self, m, arg)
def	setColumn (self, n, arg)
def	getRow (self, i)
def	getColumn (self, j)
def	swapRow (self, m1, m2)
def	swapColumn (self, n1, n2)
def	insertRow (self, i, arg)
def	insertColumn (self, j, arg)
def	extendRow (self, arg)
def	extendColumn (self, arg)
def	deleteRow (self, i)
def	deleteColumn (self, j)
def	transpose (self)
def	getBlock (self, i, j, row, column=None)
def	subMatrix (self, I, J=None)
def	toMatrix (self, flag=True)
Public Member Functions inherited from nzmath.matrix.RingMatrix
def	__add__ (self, other)
def	__sub__ (self, other)
def	__mul__ (self, other)
def	__rmul__ (self, other)
def	__mod__ (self, other)
def	__pos__ (self)
def	__neg__ (self)
def	getCoefficientRing (self)
def	toSubspace (self, isbasis=None)
def	hermiteNormalForm (self, non_zero=False)
def	exthermiteNormalForm (self, non_zero=False)
def	kernelAsModule (self)
Public Member Functions inherited from nzmath.ring.RingElement
def	__init__ (self, *args, **kwd)
def	__eq__ (self, other)
def	__hash__ (self)
def	__ne__ (self, other)

Public Attributes
	coeff_ring
	row
Public Attributes inherited from nzmath.matrix.Matrix
	row
	column
	compo
	coeff_ring
Public Attributes inherited from nzmath.matrix.RingMatrix
	row
	coeff_ring

Static Public Attributes
def	isAntisymmetricMatrix = isAlternatingMatrix
def	isSkewsymmetricMatrix = isAlternatingMatrix
def	cofactors = cofactorMatrix
def	SNF = smithNormalForm
def	elementary_divisor = smithNormalForm
def	extSNF = extsmithNormalForm
Static Public Attributes inherited from nzmath.matrix.RingMatrix
def	HNF = hermiteNormalForm
def	extHNF = exthermiteNormalForm

Private Member Functions
def	_characteristicPolyList (self)

Private Attributes
	__class__

Detailed Description

RingSquareMatrix is a class for square matrices whose elements are in ring.

Definition at line 860 of file matrix.py.

Constructor & Destructor Documentation

__init__()

def nzmath.matrix.RingSquareMatrix.__init__	(		self,
			row,
			column = 0,
			compo = 0,
			coeff_ring = 0
	)

RingSquareMatrix(row [, column ,components, coeff_ring])
RingSquareMatrix must be row == column .

Reimplemented from nzmath.matrix.SquareMatrix.

Reimplemented in nzmath.matrix.FieldSquareMatrix.

Definition at line 865 of file matrix.py.

References nzmath.matrix.Matrix._initialize().

Member Function Documentation

__pow__()

def nzmath.matrix.RingSquareMatrix.__pow__	(		self,
			other
	)

powering self to integer.

Definition at line 872 of file matrix.py.

References nzmath.matrix.Matrix.coeff_ring, nzmath.factor.misc.FactoredInteger.copy(), nzmath.imaginary.Complex.copy(), nzmath.matrix.Matrix.copy(), nzmath.quad.ReducedQuadraticForm.inverse(), nzmath.permute.Permute.inverse(), nzmath.imaginary.Complex.inverse(), nzmath.intresidue.IntegerResidueClass.inverse(), nzmath.group.GroupElement.inverse(), nzmath.permute.ExPermute.inverse(), nzmath.algfield.BasicAlgNumber.inverse(), nzmath.ring.QuotientFieldElement.inverse(), nzmath.finitefield.ExtendedFieldElement.inverse(), nzmath.module.Ideal.inverse(), nzmath.algfield.MatAlgNumber.inverse(), nzmath.rational.Integer.inverse(), nzmath.matrix.FieldSquareMatrix.inverse(), nzmath.matrix.Matrix.row, nzmath.lattice.LatticeElement.row, and nzmath.matrix.unitMatrix().

_characteristicPolyList()

def nzmath.matrix.RingSquareMatrix._characteristicPolyList (   self )

private

for characteristicPolynomial, adjugateMatrix

Assume self.row >= 2.

Definition at line 1003 of file matrix.py.

References nzmath.matrix.Matrix.coeff_ring, nzmath.matrix.Matrix.row, nzmath.lattice.LatticeElement.row, nzmath.finitefield.ExtendedFieldElement.trace(), nzmath.algfield.BasicAlgNumber.trace(), nzmath.algfield.MatAlgNumber.trace(), nzmath.matrix.RingSquareMatrix.trace(), nzmath.elliptic.ECoverGF.trace(), and nzmath.matrix.unitMatrix().

Referenced by nzmath.matrix.RingSquareMatrix.adjugateMatrix().

adjugateMatrix()

def nzmath.matrix.RingSquareMatrix.adjugateMatrix

(

self

)

Return adjugate(classical adjoint) matrix.

Definition at line 1034 of file matrix.py.

References nzmath.matrix.RingSquareMatrix._characteristicPolyList(), nzmath.matrix.Matrix.coeff_ring, nzmath.matrix.Matrix.row, nzmath.lattice.LatticeElement.row, and nzmath.matrix.unitMatrix().

Referenced by nzmath.matrix.RingSquareMatrix.cofactorMatrix().

characteristicMatrix()

def nzmath.matrix.RingSquareMatrix.characteristicMatrix

(

self

)

Return the characteristic matrix (i.e. xI-A) of self.

Definition at line 995 of file matrix.py.

References nzmath.matrix.Matrix.coeff_ring, nzmath.matrix.Matrix.row, nzmath.lattice.LatticeElement.row, and nzmath.matrix.unitMatrix().

characteristicPolynomial()

def nzmath.matrix.RingSquareMatrix.characteristicPolynomial

(

self

)

characteristicPolynomial() -> Polynomial

Definition at line 1022 of file matrix.py.

cofactor()

def nzmath.matrix.RingSquareMatrix.cofactor	(		self,
			i,
			j
	)

Return (i, j)-cofactor of self.

Definition at line 979 of file matrix.py.

References nzmath.matrix.RingSquareMatrix.determinant(), and nzmath.matrix.Matrix.subMatrix().

cofactorMatrix()

def nzmath.matrix.RingSquareMatrix.cofactorMatrix

(

self

)

Return cofactor matrix.

Definition at line 1046 of file matrix.py.

References nzmath.matrix.RingSquareMatrix.adjugateMatrix(), and nzmath.matrix.Matrix.transpose().

commutator()

def nzmath.matrix.RingSquareMatrix.commutator	(		self,
			other
	)

Return commutator defined as follows:
[self, other] = self * other - other * self .

Definition at line 988 of file matrix.py.

determinant()

def nzmath.matrix.RingSquareMatrix.determinant

(

self

)

Return determinant of self.

Reimplemented in nzmath.matrix.FieldSquareMatrix.

Definition at line 946 of file matrix.py.

References nzmath.matrix.Matrix.coeff_ring, nzmath.factor.misc.FactoredInteger.copy(), nzmath.imaginary.Complex.copy(), nzmath.matrix.Matrix.copy(), nzmath.bigrange.range(), nzmath.matrix.Matrix.row, and nzmath.lattice.LatticeElement.row.

Referenced by nzmath.matrix.RingSquareMatrix.cofactor(), and nzmath.matrix.RingSquareMatrix.isSingular().

extsmithNormalForm()

def nzmath.matrix.RingSquareMatrix.extsmithNormalForm

(

self

)

Find the Smith Normal Form M for square matrix,
Computing U,V which satisfied M=U*self*V.
Return matrices tuple,(U,V,M).

Definition at line 1116 of file matrix.py.

References nzmath.matrix.Matrix.coeff_ring, nzmath.factor.misc.FactoredInteger.copy(), nzmath.imaginary.Complex.copy(), nzmath.matrix.Matrix.copy(), nzmath.bigrange.range(), and nzmath.matrix.unitMatrix().

getRing()

def nzmath.matrix.RingSquareMatrix.getRing

(

self

)

Return matrix ring of self.

Reimplemented from nzmath.ring.RingElement.

Definition at line 904 of file matrix.py.

References nzmath.matrix.RingMatrix.getCoefficientRing(), nzmath.matrix.Matrix.row, and nzmath.lattice.LatticeElement.row.

Referenced by nzmath.poly.multiutil.RingPolynomial.__add__(), nzmath.ring.QuotientFieldElement.__add__(), nzmath.poly.uniutil.RingPolynomial.__add__(), nzmath.ring.QuotientFieldElement.__eq__(), nzmath.poly.uniutil.FieldPolynomial.__pow__(), nzmath.poly.multiutil.RingPolynomial.__radd__(), nzmath.poly.uniutil.RingPolynomial.__radd__(), nzmath.poly.multiutil.RingPolynomial.__rsub__(), nzmath.ring.QuotientFieldElement.__rsub__(), nzmath.poly.uniutil.RingPolynomial.__rsub__(), nzmath.ring.QuotientFieldElement.__rtruediv__(), nzmath.poly.multiutil.RingPolynomial.__sub__(), nzmath.ring.QuotientFieldElement.__sub__(), nzmath.poly.uniutil.RingPolynomial.__sub__(), nzmath.ring.QuotientFieldElement.__truediv__(), nzmath.poly.uniutil.PrimeCharacteristicFunctionsProvider._small_index_mod_pow(), nzmath.ring.CommutativeRingElement.exact_division(), nzmath.poly.uniutil.DivisionProvider.extgcd(), nzmath.poly.uniutil.PrimeCharacteristicFunctionsProvider.factor(), nzmath.poly.uniutil.DivisionProvider.mod_pow(), nzmath.poly.uniutil.PrimeCharacteristicFunctionsProvider.mod_pow(), nzmath.poly.uniutil.PseudoDivisionProvider.monic_pow(), nzmath.ring.CommutativeRingElement.mul_module_action(), and nzmath.poly.uniutil.SubresultantGcdProvider.subresultant_gcd().

isAlternatingMatrix()

def nzmath.matrix.RingSquareMatrix.isAlternatingMatrix

(

self

)

Check whether self is alternating matrix or not.
Alternating (skew symmetric, or antisymmetric) matrix satisfies M=-M^T.

Definition at line 917 of file matrix.py.

References nzmath.matrix.Matrix.column, nzmath.lattice.LatticeElement.column, nzmath.bigrange.range(), nzmath.matrix.Matrix.row, and nzmath.lattice.LatticeElement.row.

isOrthogonalMatrix()

def nzmath.matrix.RingSquareMatrix.isOrthogonalMatrix

(

self

)

Check whether self is orthogonal matrix or not.
Orthogonal matrix satisfies M*M^T equals unit matrix.

Definition at line 910 of file matrix.py.

References nzmath.matrix.Matrix.coeff_ring, nzmath.matrix.Matrix.row, nzmath.lattice.LatticeElement.row, nzmath.matrix.Matrix.transpose(), nzmath.factor.mpqs.Elimination.transpose(), and nzmath.matrix.unitMatrix().

isSingular()

def nzmath.matrix.RingSquareMatrix.isSingular

(

self

)

Check determinant == 0 or not.

Definition at line 931 of file matrix.py.

References nzmath.matrix.RingSquareMatrix.determinant().

smithNormalForm()

def nzmath.matrix.RingSquareMatrix.smithNormalForm

(

self

)

Find the Smith Normal Form for square non-singular integral matrix.
Return the list of diagonal elements.

Definition at line 1054 of file matrix.py.

References nzmath.matrix.Matrix.coeff_ring, nzmath.factor.misc.FactoredInteger.copy(), nzmath.imaginary.Complex.copy(), nzmath.matrix.Matrix.copy(), and nzmath.bigrange.range().

toFieldMatrix()

def nzmath.matrix.RingSquareMatrix.toFieldMatrix

(

self

)

RingSquareMatrix -> FieldSquareMatrix

Reimplemented from nzmath.matrix.RingMatrix.

Definition at line 899 of file matrix.py.

trace()

def nzmath.matrix.RingSquareMatrix.trace

(

self

)

Return trace of self.

Definition at line 937 of file matrix.py.

References nzmath.matrix.Matrix.coeff_ring, nzmath.bigrange.range(), nzmath.matrix.Matrix.row, and nzmath.lattice.LatticeElement.row.

Referenced by nzmath.matrix.RingSquareMatrix._characteristicPolyList().

Member Data Documentation

__class__

nzmath.matrix.RingSquareMatrix.__class__

private

Definition at line 901 of file matrix.py.

Referenced by nzmath.imaginary.Complex.__add__(), nzmath.vector.Vector.__add__(), nzmath.real.Real.__add__(), nzmath.poly.multivar.TermIndices.__add__(), nzmath.intresidue.IntegerResidueClass.__add__(), nzmath.module.Module.__add__(), nzmath.ring.QuotientFieldElement.__add__(), nzmath.ring.Ideal.__add__(), nzmath.ring.ResidueClass.__add__(), nzmath.poly.multivar.BasicPolynomial.__call__(), nzmath.imaginary.Complex.__div__(), nzmath.intresidue.IntegerResidueClass.__div__(), nzmath.quad.ReducedQuadraticForm.__eq__(), nzmath.real.RealField.__eq__(), nzmath.poly.multivar.TermIndices.__hash__(), nzmath.poly.ring.PolynomialRing.__hash__(), nzmath.rational.Rational.__hash__(), nzmath.poly.multivar.BasicPolynomial.__hash__(), nzmath.poly.multiutil.PolynomialRingAnonymousVariables.__hash__(), nzmath.poly.multiutil.OrderProvider.__init__(), nzmath.poly.termorder.TermOrderInterface.__init__(), nzmath.poly.uniutil.OrderProvider.__init__(), nzmath.poly.uniutil.DivisionProvider.__init__(), nzmath.poly.multiutil.RingElementProvider.__init__(), nzmath.poly.uniutil.PrimeCharacteristicFunctionsProvider.__init__(), nzmath.poly.uniutil.VariableProvider.__init__(), nzmath.poly.uniutil.RingElementProvider.__init__(), nzmath.vector.Vector.__mod__(), nzmath.quad.ReducedQuadraticForm.__mul__(), nzmath.intresidue.IntegerResidueClass.__mul__(), nzmath.vector.Vector.__mul__(), nzmath.imaginary.Complex.__mul__(), nzmath.real.Real.__mul__(), nzmath.factor.misc.FactoredInteger.__mul__(), nzmath.poly.multivar.TermIndices.__mul__(), nzmath.ring.QuotientFieldElement.__mul__(), nzmath.ring.Ideal.__mul__(), nzmath.ring.ResidueClass.__mul__(), nzmath.prime.FactoredInteger.__mul__(), nzmath.rational.Integer.__mul__(), nzmath.vector.Vector.__neg__(), nzmath.intresidue.IntegerResidueClass.__neg__(), nzmath.imaginary.Complex.__neg__(), nzmath.ring.QuotientFieldElement.__neg__(), nzmath.imaginary.Complex.__pos__(), nzmath.intresidue.IntegerResidueClass.__pos__(), nzmath.ring.ResidueClass.__pos__(), nzmath.quad.ReducedQuadraticForm.__pow__(), nzmath.factor.misc.FactoredInteger.__pow__(), nzmath.permute.Permute.__pow__(), nzmath.imaginary.Complex.__pow__(), nzmath.intresidue.IntegerResidueClass.__pow__(), nzmath.ring.QuotientFieldElement.__pow__(), nzmath.finitefield.ExtendedFieldElement.__pow__(), nzmath.prime.FactoredInteger.__pow__(), nzmath.real.Real.__radd__(), nzmath.imaginary.Complex.__rdiv__(), nzmath.intresidue.IntegerResidueClass.__rdiv__(), nzmath.algfield.NumberField.__repr__(), nzmath.poly.ratfunc.RationalFunction.__repr__(), nzmath.poly.ring.PolynomialRing.__repr__(), nzmath.real.RealField.__repr__(), nzmath.imaginary.ComplexField.__repr__(), nzmath.finitefield.FinitePrimeField.__repr__(), nzmath.module.Module.__repr__(), nzmath.poly.ring.PolynomialIdeal.__repr__(), nzmath.rational.Rational.__repr__(), nzmath.poly.multiutil.RingPolynomial.__repr__(), nzmath.poly.univar.BasicPolynomial.__repr__(), nzmath.poly.formalsum.DictFormalSum.__repr__(), nzmath.algfield.BasicAlgNumber.__repr__(), nzmath.poly.multiutil.PolynomialRingAnonymousVariables.__repr__(), nzmath.poly.ring.RationalFunctionField.__repr__(), nzmath.poly.formalsum.ListFormalSum.__repr__(), nzmath.finitefield.ExtendedFieldElement.__repr__(), nzmath.poly.multiutil.PolynomialIdeal.__repr__(), nzmath.algfield.MatAlgNumber.__repr__(), nzmath.finitefield.ExtendedField.__repr__(), nzmath.module.Ideal_with_generator.__repr__(), nzmath.poly.uniutil.RingPolynomial.__repr__(), nzmath.vector.Vector.__rmul__(), nzmath.real.Real.__rmul__(), nzmath.rational.Integer.__rmul__(), nzmath.imaginary.Complex.__rsub__(), nzmath.real.Real.__rsub__(), nzmath.intresidue.IntegerResidueClass.__rsub__(), nzmath.ring.QuotientFieldElement.__rsub__(), nzmath.real.Real.__rtruediv__(), nzmath.ring.QuotientFieldElement.__rtruediv__(), nzmath.vector.Vector.__sub__(), nzmath.imaginary.Complex.__sub__(), nzmath.real.Real.__sub__(), nzmath.poly.multivar.TermIndices.__sub__(), nzmath.intresidue.IntegerResidueClass.__sub__(), nzmath.ring.QuotientFieldElement.__sub__(), nzmath.ring.ResidueClass.__sub__(), nzmath.quad.ReducedQuadraticForm.__truediv__(), nzmath.vector.Vector.__truediv__(), nzmath.real.Real.__truediv__(), nzmath.ring.QuotientFieldElement.__truediv__(), nzmath.module.Module._module_mul(), nzmath.finitefield.ExtendedFieldElement._op(), nzmath.module.Module._rational_mul(), nzmath.module.Module._scalar_mul(), nzmath.module.Module.change_base_module(), nzmath.poly.multivar.BasicPolynomial.combine_similar_terms(), nzmath.imaginary.Complex.conjugate(), nzmath.poly.univar.PolynomialInterface.construct_with_default(), nzmath.poly.formalsum.DictFormalSum.construct_with_default(), nzmath.poly.multivar.BasicPolynomial.construct_with_default(), nzmath.poly.formalsum.ListFormalSum.construct_with_default(), nzmath.vector.Vector.copy(), nzmath.factor.misc.FactoredInteger.copy(), nzmath.imaginary.Complex.copy(), nzmath.module.Module.copy(), nzmath.prime.FactoredInteger.copy(), nzmath.module.Ideal_with_generator.copy(), nzmath.poly.multivar.BasicPolynomial.erase_variable(), nzmath.poly.multiutil.PseudoDivisionProvider.exact_division(), nzmath.poly.multivar.TermIndices.gcd(), nzmath.poly.ring.PolynomialRing.getCommonSuperring(), nzmath.poly.multiutil.PolynomialRingAnonymousVariables.getCommonSuperring(), nzmath.module.Module.intersect(), nzmath.quad.ReducedQuadraticForm.inverse(), nzmath.imaginary.Complex.inverse(), nzmath.intresidue.IntegerResidueClass.inverse(), nzmath.ring.QuotientFieldElement.inverse(), nzmath.finitefield.ExtendedFieldElement.inverse(), nzmath.module.Module.issubmodule(), nzmath.module.Module.issupermodule(), nzmath.poly.multivar.TermIndices.lcm(), nzmath.poly.termorder.TermOrderInterface.leading_coefficient(), nzmath.poly.termorder.TermOrderInterface.leading_term(), nzmath.prime.TestPrime.next(), nzmath.poly.multivar.TermIndices.pop(), nzmath.poly.univar.BasicPolynomial.square(), and nzmath.poly.multiutil.NestProvider.unnest().

coeff_ring

nzmath.matrix.RingSquareMatrix.coeff_ring

Definition at line 902 of file matrix.py.

cofactors

def nzmath.matrix.RingSquareMatrix.cofactors = cofactorMatrix

static

Definition at line 1052 of file matrix.py.

elementary_divisor

def nzmath.matrix.RingSquareMatrix.elementary_divisor = smithNormalForm

static

Definition at line 1114 of file matrix.py.

extSNF

def nzmath.matrix.RingSquareMatrix.extSNF = extsmithNormalForm

static

Definition at line 1175 of file matrix.py.

isAntisymmetricMatrix

def nzmath.matrix.RingSquareMatrix.isAntisymmetricMatrix = isAlternatingMatrix

static

Definition at line 928 of file matrix.py.

isSkewsymmetricMatrix

def nzmath.matrix.RingSquareMatrix.isSkewsymmetricMatrix = isAlternatingMatrix

static

Definition at line 929 of file matrix.py.

row

nzmath.matrix.RingSquareMatrix.row

Definition at line 1028 of file matrix.py.

SNF

def nzmath.matrix.RingSquareMatrix.SNF = smithNormalForm

static

Definition at line 1113 of file matrix.py.

---

The documentation for this class was generated from the following file:

• nzmath/matrix.py