Package sage :: Package linalg :: Module matrix :: Class Matrix_rational_dense
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Class Matrix_rational_dense

Matrix --+
         |
        Matrix_rational_dense


Method Summary
  __init__(self, parent, entries, coerce_entries, copy)
  __getitem__(self, ij)
  __mul__(self, right)
  __setitem__(self, ij, x)
  entries(self)
  hessenberg_normal_form(self)
Compute and return the Hessenberg form of self.
  list(self)
  pivots(self)
Return the i such that the i-th column of self is a pivot column of the reduced row echelon form of self.
  reduced_row_echelon_form(self)
    Inherited from Matrix
  __abs__(self)
  __add__(self, right)
  __cmp__(self, right)
  __div__(self, right)
  __invert__(self)
  __mod__(self, p)
  __neg__(self)
  __pos__(self)
  __pow__(self, n)
EXAMPLES:
  __radd__(self, right)
  __repr__(self)
  __rmul__(self, left)
  __str__(self)
  __sub__(self, right)
  add_multiple_of_column(self, i, j, s)
Replace column i by s times column j.
  add_multiple_of_row(self, i, j, s)
Replace row i by s times row j.
  augment(self, other)
Return the augmented matrix of the form [self | other].
  base_ring(self)
  block_sum(self, other)
Return the block matrix that has self and other on the diagonal: [self | 0 ] [ 0 | other ]
  change_ring(self, ring)
  charpoly(self)
  column_space(self)
  columns(self)
Returns the list of columns of self, as vectors.
  commutator(self, other)
  copy(self)
  decompose(self, is_diagonalizable)
Returns the decomposition of the vector space under the right action of self.
  decompose_subspace(self, V, is_diagonalizable)
Suppose A is a matrix acting on a vector space V, and W is a vector subspace of V.
  denominator(self)
  dense_row(self, n)
  det(self)
  determinant(self)
  echelon_form(self, *args)
Returns the reduced row echelon form INPUT: matrix -- an element A of a MatrixSpace OUTPUT: matrix -- The reduced row echelon form of A.
  fcp(self)
Return the factorization of the characteristic polynomial of self.
  get(self, ij)
  height(self)
  hermite_normal_form(self)
  image(self)
  is_dense(self)
  is_invertible(self)
  is_matrix(self)
  is_sparse(self)
  is_square(self)
  kernel(self)
ALGORITHM: Elementary row ops don't change kernel, since they are just left multiplication by an invertible matrix, so we instead compute kernel of echelon form, which is easy.
  linalg_matrix(self)
  linear_combination_of_rows(self, v)
  matrix(self, nrows, ncols, entries, coerce_entries, copy)
  matrix_space(self, nrows, ncols)
  maxspin(self, v)
Computes the largest integer n such that the list of vectors S=[v, A(v), ..., A^n(v)] are linearly independent, and returns that list.
  ncols(self)
  nonpivots(self)
Return the list of i such that the i-th column of self is NOT a pivot column of the reduced row echelon form of self.
  nonzero_positions(self)
Returns the set of pairs (i,j) such that self[i,j] != 0.
  nonzero_positions_in_column(self, i)
Return the integers j such that self[j,i] is nonzero, i.e., such that the j-th position of the i-th column is nonzero.
  nonzero_positions_in_row(self, i)
Return the integers j such that self[i,j] is nonzero, i.e., such that the j-th position of the i-th row is nonzero.
  nrows(self)
  nullity(self)
  parent(self)
  pari(self)
  rank(self)
  rational_decomposition(self)
  rational_reconstruction(self)
  rescale_row(self, i, s)
Replace i-th row of self by s times i-th row of self.
  restrict(self, V, check)
Returns the matrix that defines the action of self on the invariant subspace V.
  row(self, i)
  row_space(self)
  rows(self)
  set(self, ij, x)
  smith_normal_form(self)
  sparse_columns(self)
  sparse_rows(self)
  stack(self, other)
Return the augmented matrix self on top of other:...
  str_latex(self, hfill)
Print a latex matrix representation of self.
  str_sparse_latex(self, variable)
Return a latex string that represents this matrix as a sparse matrix.
  submatrix_from_columns(self, columns)
Return the submatrix of self of columns col[i] for i in the list of columns.
  submatrix_from_rows(self, rows)
Return the submatrix of self of rows row[i] for i in the list of rows.
  swap_columns(self, c1, c2)
Swap columns c1 and c2 of self.
  swap_rows(self, r1, r2)
Swap rows r1 and r2 of self.
  trace(self)
Return the trace of self, which is the sum of the diagonal entries of self.
  transpose(self)
Returns the transpose of self, without changing self.
  vector_matrix_multiply(self, v)
Returns the vector times matrix product.

Method Details

hessenberg_normal_form(self)

Compute and return the Hessenberg form of self. ALGORITHM: See Henri Cohen's book.
Overrides:
sage.linalg.matrix.Matrix.hessenberg_normal_form (inherited documentation)

pivots(self)

Return the i such that the i-th column of self is a
pivot column of the reduced row echelon form of self.
OUTPUT:
    list -- sorted list of integers
Overrides:
sage.linalg.matrix.Matrix.pivots (inherited documentation)

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