Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
EXAMPLES:
sage: A = matrix(GF(9,’c’), 4, [1, 1, 0,0, 0,1,0,0, 0,0,5,0, 0,0,0,5])
sage: factor(A.minpoly())
(x + 1) * (x + 2)^2
sage: A.minpoly()(A) == 0
True
sage: factor(A.charpoly())
(x + 1)^2 * (x + 2)^2
The default variable name is x, but you can specify another name:
sage: factor(A.minpoly(’y’))
(y + 1) * (y + 2)^2
We can take the minimal polynomial of symbolic matrices:
sage: t = var(’t’)
sage: m = matrix(2,[1,2,4,t])
sage: m.minimal_polynomial()
x^2 + (-t - 1)*x + t - 8
n(prec=None, digits=None)
Return a numerical approximation of self as either a real or complex number with at least the requested
number of bits or digits of precision.
INPUT:
•prec - an integer: the number of bits of precision
•digits - an integer: digits of precision
OUTPUT: A matrix coerced to a real or complex field with prec bits of precision.
EXAMPLES:
sage: d = matrix([[3, 0],[0,sqrt(2)]]) ;
sage: b = matrix([[1, -1], [2, 2]]) ; e = b * d * b.inverse();e
[ 1/2*sqrt(2) + 3/2 -1/4*sqrt(2) + 3/4]
[ -sqrt(2) + 3 1/2*sqrt(2) + 3/2]
sage: e.numerical_approx(53)
[ 2.20710678118655 0.396446609406726]
[ 1.58578643762690 2.20710678118655]
sage: e.numerical_approx(20)
[ 2.2071 0.39645]
[ 1.5858 2.2071]
sage: (e-I).numerical_approx(20)
[2.2071 - 1.0000*I 0.39645]
[ 1.5858 2.2071 - 1.0000*I]
sage: M=matrix(QQ,4,[i/(i+1) for i in range(12)]);M
[ 0 1/2 2/3]
[ 3/4 4/5 5/6]
[ 6/7 7/8 8/9]
[ 9/10 10/11 11/12]
sage: M.numerical_approx()
[0.000000000000000 0.500000000000000 0.666666666666667]
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