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 sage: factor(a.charpoly()) x^2 * (x^2 - 34*x - 80) sage: b = matrix(QQ, 4, [-1, 2, 2, 0, 0, 4, 2, 2, 0, 0, -1, -2, 0, -4, 0, 4]) sage: a = matrix(QQ, 4, [1, 1, 0,0, 0,1,0,0, 0,0,5,0, 0,0,0,5]) sage: c = b^(-1)*a*b sage: factor(c.minpoly()) (x - 5) * (x - 1)^2 sage: factor(c.charpoly()) (x - 5)^2 * (x - 1)^2 prod_of_row_sums(cols) randomize(density=1, num_bound=2, den_bound=2, distribution=None, nonzero=False) Randomize density proportion of the entries of this matrix, leaving the rest unchanged. If x and y are given, randomized entries of this matrix have numerators and denominators bounded by x and y and have density 1. INPUT: •density - number between 0 and 1 (default: 1) •num_bound - numerator bound (default: 2) •den_bound - denominator bound (default: 2) •distribution - None or ‘1/n’ (default: None); if ‘1/n’ then num_bound, den_bound are ignored and numbers are chosen using the GMP function mpq_randomize_entry_recip_uniform •nonzero - Bool (default: False); whether the new entries are forced to be non-zero OUTPUT: •None, the matrix is modified in-space EXAMPLES: sage: a = matrix(QQ,2,4); a.randomize(); a [ 0 -1 2 -2] [ 1 -1 2 1] sage: a = matrix(QQ,2,4); a.randomize(density=0.5); a [ -1 -2 0 0] [ 0 0 1/2 0] sage: a = matrix(QQ,2,4); a.randomize(num_bound=100, den_bound=100); a [ 14/27 21/25 43/42 -48/67] [-19/55 64/67 -11/51 76] sage: a = matrix(QQ,2,4); a.randomize(distribution=’1/n’); a [ 3 1/9 1/2 1/4] [ 1 1/39 2 -1955/2] rank() Return the rank of this matrix. EXAMPLES:: sage: matrix(QQ,3,[1..9]).rank() 2 sage: matrix(QQ,100,[1..100^2]).rank() 2 row(i, from_list=False) Return the i-th row of this matrix as a dense vector. INPUT: • i - integer • from_list - ignored 346 Chapter 18. Dense matrices over the rational field
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 EXAMPLES: sage: matrix(QQ,2,[1/5,-2/3,3/4,4/9]).row(1) (3/4, 4/9) sage: matrix(QQ,2,[1/5,-2/3,3/4,4/9]).row(1,from_list=True) (3/4, 4/9) sage: matrix(QQ,2,[1/5,-2/3,3/4,4/9]).row(-2) (1/5, -2/3) set_row_to_multiple_of_row(i, j, s) Set row i equal to s times row j. EXAMPLES: sage: a = matrix(QQ,2,3,range(6)); a [0 1 2] [3 4 5] sage: a.set_row_to_multiple_of_row(1,0,-3) sage: a [ 0 1 2] [ 0 -3 -6] transpose() Returns the transpose of self, without changing self. EXAMPLES: We create a matrix, compute its transpose, and note that the original matrix is not changed. sage: A = matrix(QQ,2,3,xrange(6)) sage: type(A) sage: B = A.transpose() sage: print B [0 3] [1 4] [2 5] sage: print A [0 1 2] [3 4 5] .T is a convenient shortcut for the transpose: sage: print A.T [0 3] [1 4] [2 5] sage: A.subdivide(None, 1); A [0|1 2] [3|4 5] sage: A.transpose() [0 3] [---] [1 4] [2 5] 347
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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