[next][previous][up][top][index]
search for:
Matrix -- the class of all matrices

The type Matrix is a member of the class Type. Each object of class Matrix is called a matrix. Each matrix is also a member of class ModuleMap.

More general types (whose methods may also apply) :

  • ModuleMap -- the class of all maps between modules
  • HashTable -- the class of all hash tables

    • A matrix is a homomorphism between two modules, together with an integer (or vector of integers) called its degree, which is used when determining whether the map is homogeneous. The matrix is stored in the usual way as a rectangular array of ring elements. When the source or target modules are not free, the matrix is interpreted as a linear transformation in terms of the generators of the modules.

    See also:

  • matrices

    • A matrix f is an immutable object, so if you want to cache information about it, put it in the hash table f.cache.

    Common ways to make a matrix:

  • map -- make a map
  • matrix -- make a matrix

    • Common ways to get information about matrices:

  • degree Matrix -- the degree
  • isHomogeneous Matrix -- test for homogeneity
  • matrix Matrix -- make a matrix

    • Common operations on matrices:

  • Matrix + Matrix -- a binary operator
  • Matrix - Matrix -- a unary or binary operator, usually used for negation or subtraction
  • RingElement * Matrix -- a binary operator, usually used for multiplication
  • Matrix * Matrix -- matrix multiplication
  • Matrix == Matrix -- equality
  • Matrix ++ Matrix -- direct sum of maps
  • Matrix ** Matrix -- tensor product of matrices
  • Matrix % Matrix -- find the normal form modulo the image of a map
  • Matrix // Matrix -- factor a map through another
  • Matrix | Matrix -- join matrices horizontally
  • Matrix || Matrix -- join matrices vertically
  • Matrix ^ {...} -- select some rows of a matrix
  • Matrix _ {...} -- get some columns from a matrix

    • Common ways to use a matrix:

  • cokernel Matrix -- cokernel of a map
  • image Matrix -- image of a map
  • kernel Matrix -- kernel of a map
  • homology(Matrix,Matrix) -- kernel modulo image

    • Functions and methods returning a a matrix :

  • - Matrix
  • adjoint(Matrix,Module,Module)
  • adjoint1(Matrix,Module,Module)
  • ambient Matrix
  • basis Ideal
  • basis Module -- basis of a module
  • basis Ring -- basis of a ring
  • basis(List,Ideal)
  • basis({...},Module) -- basis of the part of a module of a certain degree
  • basis(List,Ring)
  • basis(ZZ,Ideal)
  • basis(ZZ,Module)
  • basis(ZZ,Ring)
  • borel Matrix
  • ChainComplexMap _ ZZ -- get component
  • complement Matrix
  • compress Matrix
  • contract'(Matrix,Matrix)
  • contract(Matrix,Matrix) -- contract a matrix by a matrix
  • cover Matrix
  • coverMap Module
  • diff'(Matrix,Matrix)
  • diff(Matrix,Matrix) -- differentiate a matrix by a matrix
  • divideByVariable(Matrix,RingElement)
  • divideByVariable(Matrix,RingElement,ZZ)
  • dual Matrix -- dual of a map
  • Ext^ZZ(Matrix,Module)
  • Ext^ZZ(Module,Matrix)
  • exteriorPower(ZZ,Matrix) -- exterior power
  • flatten Matrix
  • flip(Module,Module) -- matrix of commutativity of tensor product
  • generators GroebnerBasis -- the Groebner basis matrix
  • generators Ideal
  • generators Module -- matrix of generators
  • generators(ZZ,Resolution)
  • getChangeMatrix GroebnerBasis
  • getChangeMatrix(ZZ,Resolution)
  • GradedModuleMap _ ZZ
  • HH^ZZ ChainComplexMap -- cohomology of a chain complex map
  • HH_ZZ ChainComplexMap -- homology of a chain complex map
  • Hom(Matrix,Module)
  • Hom(Module,Matrix)
  • homogenize(Matrix,RingElement)
  • homogenize(Matrix,RingElement,List)
  • homomorphism Matrix
  • inducedMap(Module,Module)
  • inducedMap(Module,Module,Matrix)
  • jacobian Ideal
  • jacobian Matrix
  • jacobian MonomialIdeal
  • jacobian Ring
  • koszul(ZZ,Matrix) -- a differential in a Koszul complex
  • leadTerm Ideal
  • leadTerm Matrix
  • leadTerm(ZZ,Ideal)
  • leadTerm(ZZ,Matrix)
  • leadTerm(ZZ,ZZ,Resolution)
  • lift(Matrix,Ring)
  • map Ideal
  • map Matrix -- make a map
  • map Module -- make a map
  • map(Ideal,Ideal)
  • map(Module,Matrix) -- make a map
  • map(Module,Module) -- make a map
  • map(Module,Module,{...}) -- make a map
  • map(Module,Module,Function) -- make a map
  • map(Module,Module,Matrix) -- make a map
  • map(Module,Module,RingElement) -- make a map
  • map(Module,Module,ZZ) -- make a map
  • map(Module,Nothing,{...}) -- make a map
  • map(Module,Nothing,Matrix)
  • map(Module,RingElement) -- make a map
  • map(Module,ZZ)
  • map(Module,ZZ,{...}) -- make a map
  • map(Module,ZZ,Function) -- make a map
  • Matrix % GroebnerBasis
  • Matrix % Ideal
  • Matrix % Matrix -- find the normal form modulo the image of a map
  • Matrix % Module
  • Matrix % MonomialIdeal
  • Matrix * Matrix -- matrix multiplication
  • Matrix ** Matrix -- tensor product of matrices
  • Matrix ** Module -- tensor product
  • Matrix ** Ring -- tensor product
  • Matrix + Matrix
  • Matrix ++ Matrix -- direct sum of maps
  • Matrix ++ RingElement
  • Matrix - Matrix
  • Matrix // GroebnerBasis
  • Matrix // Matrix -- factor a map through another
  • Matrix // MonomialIdeal
  • Matrix ^ [...] -- select some rows of blocks
  • Matrix ^ {...} -- select some rows of a matrix
  • Matrix ^ ZZ
  • Matrix _ [...] -- select some columns of blocks
  • Matrix _ {...} -- get some columns from a matrix
  • matrix {...} -- make a matrix
  • Matrix | Matrix -- join matrices horizontally
  • Matrix || Matrix -- join matrices vertically
  • matrix Matrix -- make a matrix
  • matrix(Ring,{...}) -- make a matrix
  • mingens GroebnerBasis -- a matrix whose columns are minimal generators of the submodule
  • mingens Ideal
  • mingens Module
  • mingens(ZZ,Resolution)
  • Module ** Matrix
  • Module ^ [...] -- projection onto some factors of a direct sum module
  • Module ^ {...} -- projection map from a free module
  • Module _ [...] -- get inclusion map into direct sum
  • Module _ {...} -- map from free module to some generators
  • modulo(Matrix,Matrix)
  • modulo(Matrix,Nothing)
  • modulo(Nothing,Matrix)
  • presentation Module
  • presentation QuotientRing
  • prune Matrix -- minimize generators and relations
  • random(Module,Module) -- make a random module map
  • relations Module
  • reshape(Module,Module,Matrix)
  • RingElement ++ Matrix
  • RingElement ++ RingElement
  • RingElement // GroebnerBasis
  • RingElement | RingElement
  • RingElement || RingElement
  • RingMap Matrix
  • selectInSubring(ZZ,Matrix)
  • submatrix(Matrix,Sequence,Sequence)
  • submatrix(Matrix,Sequence,List)
  • submatrix(Matrix,List)
  • submatrix(Matrix,List,Sequence)
  • submatrix(Matrix,List,List)
  • substitute(Matrix,List)
  • substitute(Matrix,Matrix)
  • substitute(Matrix,Ring)
  • substitute(Matrix,ZZ)
  • sum ChainComplexMap -- direct sum of the components of a chain map
  • super Matrix
  • symmetricPower(ZZ,Matrix) -- symmetric power
  • syz GroebnerBasis -- retrieve the syzygy matrix
  • syz Matrix -- compute the syzygy matrix
  • tensorAssociativity(Module,Module,Module)
  • transpose Matrix
  • vars Ring -- row matrix of the variables
  • wedgeProduct(ZZ,ZZ,Module)

    • Methods for using a matrix :

  • autoReduce Matrix
  • betti Matrix -- display of the degrees of a map
  • chainComplex Matrix -- make a small chain complex
  • coefficients Matrix
  • coefficients(List,Matrix)
  • coefficients(ZZ,Matrix)
  • coimage Matrix
  • cokernel Matrix
  • components Matrix
  • content Matrix
  • contract(Matrix,RingElement)
  • contract(Matrix,Vector)
  • contract(RingElement,Matrix)
  • contract(Vector,Matrix)
  • degree Matrix
  • degrees Matrix -- degrees of generators
  • det Matrix -- determinant of a matrix
  • diff Matrix -- differentiate a matrix
  • diff(Matrix,RingElement)
  • diff(Matrix,Vector)
  • diff(RingElement,Matrix)
  • diff(Vector,Matrix)
  • entries Matrix
  • expression Matrix
  • Ext^ZZ(Matrix,Ideal)
  • Ext^ZZ(Matrix,Ring)
  • extend(ChainComplex,ChainComplex,Matrix)
  • forceGB Matrix
  • gb Matrix
  • homology(Matrix,Matrix) -- kernel modulo image
  • ideal Matrix
  • image Matrix
  • inducedMap(Module,Nothing,Matrix)
  • inducedMap(Nothing,Module,Matrix)
  • inducedMap(Nothing,Nothing,Matrix)
  • inducesWellDefinedMap(Module,Module,Matrix)
  • inducesWellDefinedMap(Module,Nothing,Matrix)
  • inducesWellDefinedMap(Nothing,Module,Matrix)
  • inducesWellDefinedMap(Nothing,Nothing,Matrix)
  • isHomogeneous Matrix -- test for homogeneity
  • isInjective Matrix
  • isIsomorphism Matrix
  • isSurjective Matrix
  • isWellDefined Matrix
  • kernel Matrix
  • lift(Matrix,ZZ)
  • map(Ring,Matrix)
  • map(Ring,Ring,Matrix) -- make a ring map
  • Matrix % RingElement -- reduce the columns modulo of a ring element
  • Matrix * QQ
  • Matrix * Vector
  • Matrix * ZZ
  • Matrix ** QQ
  • Matrix ** RingElement
  • Matrix ** ZZ
  • Matrix + RingElement
  • Matrix + ZZ
  • Matrix ++ ZZ
  • Matrix - RingElement
  • Matrix - ZZ
  • Matrix // RingElement -- factor a map through a multiplication map
  • Matrix // ZZ
  • Matrix == Matrix
  • Matrix == RingElement
  • Matrix == ZZ
  • Matrix _ (...) -- get an entry
  • Matrix _ ZZ -- get a column from a matrix
  • Matrix | RingElement
  • Matrix | ZZ
  • Matrix || RingElement
  • Matrix || ZZ
  • minors(ZZ,Matrix) -- ideal generated by minors
  • monomialIdeal Matrix -- make a monomial ideal
  • net Matrix
  • newCoordinateSystem(PolynomialRing,Matrix)
  • oldCoefficients Matrix
  • oldCoefficients(List,Matrix)
  • pfaffians(ZZ,Matrix)
  • precedence Matrix
  • promote(Matrix,QQ)
  • promote(Matrix,ZZ)
  • QQ * Matrix
  • QQ ** Matrix
  • rank Matrix
  • resolution Matrix -- make a projective resolution
  • ring Matrix
  • RingElement % Matrix
  • RingElement * Matrix
  • RingElement ** Matrix
  • RingElement + Matrix
  • RingElement - Matrix
  • RingElement // Matrix
  • RingElement == Matrix
  • RingElement | Matrix
  • RingElement || Matrix
  • sortColumns Matrix
  • submatrix(Matrix,List,Nothing)
  • submatrix(Matrix,Nothing,List)
  • subquotient(Matrix,Matrix)
  • subquotient(Matrix,Nothing)
  • subquotient(Nothing,Matrix)
  • substitute(Ideal,Matrix)
  • substitute(Matrix,Option)
  • substitute(Module,Matrix)
  • substitute(RingElement,Matrix)
  • substitute(Vector,Matrix)
  • toExternalString Matrix
  • topCoefficients Matrix
  • toString Matrix
  • trace Matrix
  • ZZ * Matrix
  • ZZ ** Matrix
  • ZZ + Matrix
  • ZZ ++ Matrix
  • ZZ - Matrix
  • ZZ // Matrix
  • ZZ == Matrix
  • ZZ | Matrix
  • ZZ || Matrix

    • [next][previous][up][top][index]
    search for: