Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 [ 6 7 8] sage: X.insert_row(3, [1,5,-10]) [ 0 1 2] [ 3 4 5] [ 6 7 8] [ 1 5 -10] is_LLL_reduced(delta=None, eta=None) Return True if this lattice is (δ, η)-LLL reduced. See self.LLL for a definition of LLL reduction. INPUT: •delta - parameter as described above (default: 3/4) •eta - parameter as described above (default: 0.501) EXAMPLE: sage: A = random_matrix(ZZ, 10, 10) sage: L = A.LLL() sage: A.is_LLL_reduced() False sage: L.is_LLL_reduced() True minpoly(var=’x’, algorithm=’linbox’) INPUT: •var - a variable name •algorithm - ‘linbox’ (default) ‘generic’ Note: Linbox charpoly disabled on 64-bit machines, since it hangs in many cases. EXAMPLES: sage: A = matrix(ZZ,6, range(36)) sage: A.minpoly() x^3 - 105*x^2 - 630*x sage: n=6; A = Mat(ZZ,n)([k^2 for k in range(n^2)]) sage: A.minpoly() x^4 - 2695*x^3 - 257964*x^2 + 1693440*x pivots() Return the pivot column positions of this matrix. OUTPUT: a tuple of Python integers: the position of the first nonzero entry in each row of the echelon form. EXAMPLES: sage: n = 3; A = matrix(ZZ,n,range(n^2)); A [0 1 2] [3 4 5] [6 7 8] sage: A.pivots() (0, 1) sage: A.echelon_form() [ 3 0 -3] [ 0 1 2] [ 0 0 0] 330 Chapter 17. Dense matrices over the integer ring
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 prod_of_row_sums(cols) Return the product of the sums of the entries in the submatrix of self with given columns. INPUT: •cols - a list (or set) of integers representing columns of self. OUTPUT: an integer EXAMPLES: sage: a = matrix(ZZ,2,3,[1..6]); a [1 2 3] [4 5 6] sage: a.prod_of_row_sums([0,2]) 40 sage: (1+3)*(4+6) 40 sage: a.prod_of_row_sums(set([0,2])) 40 randomize(density=1, x=None, y=None, distribution=None, nonzero=False) Randomize density proportion of the entries of this matrix, leaving the rest unchanged. The parameters are the same as the ones for the integer ring’s random_element function. If x and y are given, randomized entries of this matrix have to be between x and y and have density 1. INPUT: •self - a mutable matrix over ZZ •density - a float between 0 and 1 •x, y - if not None, these are passed to the ZZ.random_element function as the upper and lower endpoints in the uniform distribution •distribution - would also be passed into ZZ.random_element if given •nonzero - bool (default: False); whether the new entries are guaranteed to be zero OUTPUT: •None, the matrix is modified in-place EXAMPLES: sage: A = matrix(ZZ, 2,3, [1..6]); A [1 2 3] [4 5 6] sage: A.randomize() sage: A [-8 2 0] [ 0 1 -1] sage: A.randomize(x=-30,y=30) sage: A [ 5 -19 24] [ 24 23 -9] rank() Return the rank of this matrix. OUTPUT: •nonnegative integer - the rank 331
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Sage Reference Manual: Matrices and
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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