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 0| 0 -4]
[ 0 0| 0 0| 0 0| 1 4]
sage: E = matrix(QQ, [[ 0, -8, 4, -6, -2, 5, -3, 11],
... [-2, -4, 2, -4, -2, 4, -2, 6],
... [ 5, 14, -7, 12, 3, -8, 6, -27],
... [-3, -8, 7, -5, 0, 2, -6, 17],
... [ 0, 5, 0, 2, 4, -4, 1, 2],
... [-3, -7, 5, -6, -1, 5, -4, 14],
... [ 6, 18, -10, 14, 4, -10, 10, -28],
... [-2, -6, 4, -5, -1, 3, -3, 13]])
sage: E.minimal_polynomial().factor()
(x - 2)^3
sage: E.characteristic_polynomial().factor()
(x - 2)^8
sage: E.rational_form()
[ 2| 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 -4| 0 0 0]
[ 0| 0 0| 1 4| 0 0 0]
[---+-------+-------+-----------]
[ 0| 0 0| 0 0| 0 0 8]
[ 0| 0 0| 0 0| 1 0 -12]
[ 0| 0 0| 0 0| 0 1 6]
The principal feature of rational canonical form is that it can be computed over any field using only field
operations. Other forms, such as Jordan canonical form, are complicated by the need to determine the
eigenvalues of the matrix, which can lie outside the field. The following matrix has all of its eigenvalues
outside the rationals - some are irrational (± √ 2) and the rest are complex (−1 ± 2i).
sage: A = matrix(QQ,
... [[-154, -3, -54, 44, 48, -244, -19, 67, -326, 85, 355, 581],
... [ 504, 25, 156, -145, -171, 793, 99, -213, 1036, -247, -1152, -1865],
... [ 294, -1, 112, -89, -90, 469, 36, -128, 634, -160, -695, -1126],
... [ -49, -32, 25, 7, 37, -64, -58, 12, -42, -14, 72, 106],
... [-261, -123, 65, 47, 169, -358, -254, 70, -309, -29, 454, 673],
... [-448, -123, -10, 109, 227, -668, -262, 163, -721, 95, 896, 1410],
... [ 38, 7, 8, -14, -17, 66, 6, -23, 73, -29, -78, -143],
... [ -96, 10, -55, 37, 24, -168, 17, 56, -231, 88, 237, 412],
... [ 310, 67, 31, -81, -143, 473, 143, -122, 538, -98, -641, -1029],
... [ 139, -35, 99, -49, -18, 236, -41, -70, 370, -118, -377, -619],
... [ 243, 9, 81, -72, -81, 386, 43, -105, 508, -124, -564, -911],
... [-155, -3, -55, 45, 50, -245, -27, 65, -328, 77, 365, 583]])
sage: A.characteristic_polynomial().factor()
(x^2 - 2)^2 * (x^2 + 2*x + 5)^4
sage: A.eigenvalues(extend=False)
[]
sage: A.rational_form()
[ 0 -5| 0 0 0 0| 0 0 0 0 0 0]
[ 1 -2| 0 0 0 0| 0 0 0 0 0 0]
[-------+---------------+-----------------------]
[ 0 0| 0 0 0 10| 0 0 0 0 0 0]
[ 0 0| 1 0 0 4| 0 0 0 0 0 0]
[ 0 0| 0 1 0 -3| 0 0 0 0 0 0]
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