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Package sage :: Package linalg :: Module matrix :: Class Matrix_rational_dense |
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Matrix
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Matrix_rational_dense
Method Summary | |
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__init__(self,
parent,
entries,
coerce_entries,
copy)
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__getitem__(self,
ij)
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__mul__(self,
right)
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__setitem__(self,
ij,
x)
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entries(self)
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Compute and return the Hessenberg form of self. | |
list(self)
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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)
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Inherited from Matrix | |
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EXAMPLES: | |
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Replace column i by s times column j. | |
Replace row i by s times row j. | |
Return the augmented matrix of the form [self | other]. | |
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Return the block matrix that has self and other on the diagonal: [self | 0 ] [ 0 | other ] | |
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Returns the list of columns of self, as vectors. | |
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Returns the decomposition of the vector space under the right action of self. | |
Suppose A is a matrix acting on a vector space V, and W is a vector subspace of V. | |
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Returns the reduced row echelon form INPUT: matrix -- an element A of a MatrixSpace OUTPUT: matrix -- The reduced row echelon form of A. | |
Return the factorization of the characteristic polynomial of self. | |
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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. | |
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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. | |
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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. | |
Returns the set of pairs (i,j) such that self[i,j] != 0. | |
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. | |
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. | |
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Replace i-th row of self by s times i-th row of self. | |
Returns the matrix that defines the action of self on the invariant subspace V. | |
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Return the augmented matrix self on top of other:... | |
Print a latex matrix representation of self. | |
Return a latex string that represents this matrix as a sparse matrix. | |
Return the submatrix of self of columns col[i] for i in the list of columns. | |
Return the submatrix of self of rows row[i] for i in the list of rows. | |
Swap columns c1 and c2 of self. | |
Swap rows r1 and r2 of self. | |
Return the trace of self, which is the sum of the diagonal entries of self. | |
Returns the transpose of self, without changing self. | |
Returns the vector times matrix product. |
Method Details |
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hessenberg_normal_form(self)Compute and return the Hessenberg form of self. ALGORITHM: See Henri Cohen's book.
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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
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