Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[-+-+-+-+-+-+-+-]
[0|0|0|0|0|1|2|1]
[-+-+-+-+-+-+-+-]
[0|0|0|0|0|0|0|2]
sage: U.inverse()*C*U == Z
True
sage: D = matrix(QQ, [[ -4, 3, 7, 2, -4, 5, 7, -3],
... [ -6, 5, 7, 2, -4, 5, 7, -3],
... [ 21, -12, 89, 25, 8, 27, 98, -95],
... [ -9, 5, -44, -11, -3, -13, -48, 47],
... [ 23, -13, 74, 21, 12, 22, 85, -84],
... [ 31, -18, 135, 38, 12, 47, 155, -147],
... [-33, 19, -138, -39, -13, -45, -156, 151],
... [ -7, 4, -29, -8, -3, -10, -34, 34]])
sage: Z, U = D.zigzag_form(transformation=True)
sage: Z
[ 0 -4| 0 0| 0 0| 0 0]
[ 1 4| 0 0| 0 0| 0 0]
[-----+-----+-----+-----]
[ 0 0| 0 1| 0 0| 0 0]
[ 0 0|-4 4| 0 0| 0 0]
[-----+-----+-----+-----]
[ 0 0| 0 0| 0 -4| 0 0]
[ 0 0| 0 0| 1 4| 0 0]
[-----+-----+-----+-----]
[ 0 0| 0 0| 0 0| 0 1]
[ 0 0| 0 0| 0 0|-4 4]
sage: U.inverse()*D*U == Z
True
sage: C.jordan_form() == D.jordan_form()
True
ZigZag form is achieved entirely with the operations of the field, so while the eigenvalues may lie outside
the field, this does not impede the computation of the form.
sage: F. = GF(5^4)
sage: A = matrix(F, [[ a, 0, 0, a + 3],
... [ 0,a^2 + 1, 0, 0],
... [ 0, 0,a^3, 0],
... [a^2 +4 , 0, 0,a + 2]])
sage: A.zigzag_form()
[ 0 a^3 + 2*a^2 + 2*a + 2| 0| 0]
[ 1 2*a + 2| 0| 0]
[-------------------------------------------+---------------------+---------------------]
[ 0 0| a^3| 0]
[-------------------------------------------+---------------------+---------------------]
[ 0 0| 0| a^2 + 1]
sage: A.eigenvalues()
Traceback (most recent call last):
...
Subdivisions are optional.
sage: F. = GF(5^4)
sage: A = matrix(F, [[ a, 0, 0, a + 3],
... [ 0,a^2 + 1, 0, 0],
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