Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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