Matrix

Matrix -- the class of all matrices for which Groebner basis operations are available from the engine.

A matrix is a map from a graded module to a graded module, see Module. The degree of the map is not necessarily 0, and may be obtained with degree.

Multiplication of matrices corresponds to composition of maps, and when f and g are maps so that the target Q of g equals the source P of f, the product f*g is defined, its source is the source of g, and its target is the target of f. The degree of f*g is the sum of the degrees of f and of g. The product is also defined when P != Q, provided only that P and Q are free modules of the same rank. If the degrees of P differ from the corresponding degrees of Q by the same degree d, then the degree of f*g is adjusted by d so it will have a good chance to be homogeneous, and the target and source of f*g are as before.

If h is a matrix then h_j is the j-th column of the matrix, and h_j_i is the entry in row i, column j. The notation h_(i,j) can be used as an abbreviation for h_j_i, allowing row and column indices to be written in the customary order.

If m and n are matrices, a is a ring element, and i is an integer, then m+n, m-n, -m, m n, a*m, and i*m denote the usual matrix arithmetic. Use m == n, and m == 0 to check equality of matrices.

Operations which produce matrices:

  • flip
  • genericMatrix -- an generic matrix
  • genericSkewMatrix -- an generic skew-symmetric matrix
  • genericSymmetricMatrix -- a generic symmetric matrix
  • id -- identity maps
  • matrix -- create a matrix
  • map -- create a map of modules
  • random -- a random homgeneous matrix

    • Operations on matrices:

  • == -- equality test
  • != -- inequality test
  • + -- sum
  • - -- difference
  • * -- product
  • ^ -- power
  • Matrix ^ List -- extracting or permuting rows
  • Matrix ^ Array -- extracting or permuting blocks of rows
  • Matrix % Matrix -- remainder
  • Matrix % RingElement -- remainder
  • Matrix // Matrix -- quotient
  • Matrix // RingElement -- quotient
  • Matrix _ Sequence -- getting an entry
  • Matrix _ List -- extracting or permuting columns
  • Matrix _ Array -- extracting or permuting blocks of columns
  • Matrix | Matrix -- horizontal concatenation
  • Matrix || Matrix -- vertical concatenation
  • Matrix ++ Matrix -- direct sum
  • Matrix ** Matrix -- tensor product of matrices
  • Matrix ** Module -- tensor product, e.g., degree shifting
  • Matrix ** Ring -- tensor product, base change
  • :
  • adjoint
  • adjoint1
  • ambient
  • basis -- k-basis of a module in a given degree
  • borel m -- smallest Borel submodule containing lead monomials of m
  • codim
  • complement
  • compress -- removal of zero columns
  • content
  • contract(m,n) -- contraction of n by m (differentiation without the coefficients)
  • degree
  • det -- determinant
  • diff -- differentiation
  • divideByVariable -- divide columns by a variable repeatedly
  • dual
  • selectInSubring
  • entries -- the entries of m
  • exteriorPower -- exterior power of m
  • flatten -- collect entries of a matrix into one row
  • inducedMap -- a map induced on subquotients
  • inducesWellDefinedMap -- whether a matrix would induce a well defined map
  • isHomogeneous -- whether a matrix is homogeneous
  • isInjective -- whether a map is injective
  • isIsomorphism -- whether a map is an isomorphism
  • isSurjective -- whether a map is surjective
  • isWellDefined -- whether a map is well-defined
  • homogenize -- homogenize a matrix
  • jacobian -- Jacobian matrix of a matrix
  • koszul -- i-th Koszul matrix of a matrix
  • leadTerm -- lead monomial matrix of the columns of a matrix
  • minors -- ideal minors of a matrix
  • modulo
  • pfaffians -- ideal of i by i Pfaffians of a skew symmetric matrix
  • poincare
  • reshape
  • ring -- the base ring of a matrix
  • singularLocus
  • sortColumns -- sort the columns of a matrix
  • source -- the source free module of a map
  • submatrix -- extract a submatrix
  • substitute -- replacing the variables in a matrix
  • symmetricPower -- symmetric power of a matrix
  • target -- the target module of a map
  • top
  • topCoefficients
  • trace -- trace
  • transpose -- transpose a matrix

    • Operations which produce modules from matrices:

  • cokernel -- the cokernel of a map
  • homology
  • image
  • kernel
  • kernel -- the kernel of a map
  • submodule
  • subquotient

    • Operations which produce Groebner bases from matrices:

  • gb
  • mingens
  • syz

    • Printing matrices:

  • compactMatrixForm

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