Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
In this example we take the symbolic answer and make it numerical at the end:
sage: exp(matrix(SR, [[1.2, 5.6], [3,4]])).change_ring(RDF)
[346.557487298 661.734590934]
[354.500673715 677.424782765]
Another example involving the reversed identity matrix, which we clumsily create:
sage: m = identity_matrix(SR,4); m = matrix(list(reversed(m.rows()))) * x
sage: exp(m)
[1/2*(e^(2*x) + 1)*e^(-x) 0 0 1/2*(e^(2*x) - 1
[ 0 1/2*(e^(2*x) + 1)*e^(-x) 1/2*(e^(2*x) - 1)*e^(-x)
[ 0 1/2*(e^(2*x) - 1)*e^(-x) 1/2*(e^(2*x) + 1)*e^(-x)
[1/2*(e^(2*x) - 1)*e^(-x) 0 0 1/2*(e^(2*x) + 1
expand()
Operates point-wise on each element.
EXAMPLES:
sage: M = matrix(2, 2, range(4)) - var(’x’)
sage: M*M
[ x^2 + 2 -2*x + 3]
[ -4*x + 6 (x - 3)^2 + 2]
sage: (M*M).expand()
[ x^2 + 2 -2*x + 3]
[ -4*x + 6 x^2 - 6*x + 11]
factor()
Operates point-wise on each element.
EXAMPLES:
sage: M = matrix(SR, 2, 2, x^2 - 2*x + 1); M
[x^2 - 2*x + 1 0]
[ 0 x^2 - 2*x + 1]
sage: M.factor()
[(x - 1)^2 0]
[ 0 (x - 1)^2]
fcp(var=’x’)
Return the factorization of the characteristic polynomial of self.
INPUT:
•var - (default: ‘x’) name of variable of charpoly
EXAMPLES:
sage: a = matrix(SR,[[1,2],[3,4]])
sage: a.fcp()
x^2 - 5*x - 2
sage: [i for i in a.fcp()]
[(x^2 - 5*x - 2, 1)]
sage: a = matrix(SR,[[1,0],[0,2]])
sage: a.fcp()
(x - 2) * (x - 1)
sage: [i for i in a.fcp()]
[(x - 2, 1), (x - 1, 1)]
sage: a = matrix(SR, 5, [1..5^2])
sage: a.fcp()
(x^2 - 65*x - 250) * x^3
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