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|-25 10| 0 0]
[---------------+-------+-------]
[ 0 0 0 0| 1 0| 0 -4]
[ 0 0 0 0| 0 0| 1 -4]
sage: U.inverse()*A*U == Z
True
sage: B = matrix(QQ, [[ 16, 69, -13, 2, -52, 143, 90, -3],
... [ 26, 54, 6, -5, -28, 73, 73, -48],
... [-16, -79, 12, -10, 64, -142, -115, 41],
... [ 27, -7, 21, -33, 39, -20, -42, 43],
... [ 8, -75, 34, -32, 86, -156, -130, 42],
... [ 2, -17, 7, -8, 20, -33, -31, 16],
... [-24, -80, 7, -3, 56, -136, -112, 42],
... [ -6, -19, 0, -1, 13, -28, -27, 15]])
sage: Z, U = B.zigzag_form(transformation=True)
sage: Z
[ 0 0 0 0 0 1000| 0| 0]
[ 1 0 0 0 0 900| 0| 0]
[ 0 1 0 0 0 -30| 0| 0]
[ 0 0 1 0 0 -153| 0| 0]
[ 0 0 0 1 0 3| 0| 0]
[ 0 0 0 0 1 9| 0| 0]
[-----------------------------+----+----]
[ 0 0 0 0 0 0| -2| 0]
[-----------------------------+----+----]
[ 0 0 0 0 0 0| 1| -2]
sage: U.inverse()*B*U == Z
True
sage: A.jordan_form() == B.jordan_form()
True
Two more examples, illustrating the two extremes of the zig-zag nature of this form. The first has a one in
each of the off-diagonal blocks, the second has all zeros in each off-diagonal block. Notice again that the
two matrices are similar, since their Jordan canonical forms are equal.
sage: C = matrix(QQ, [[2, 31, -10, -9, -125, 13, 62, -12],
... [0, 48, -16, -16, -188, 20, 92, -16],
... [0, 9, -1, 2, -33, 5, 18, 0],
... [0, 15, -5, 0, -59, 7, 30, -4],
... [0, -21, 7, 2, 84, -10, -42, 5],
... [0, -42, 14, 8, 167, -17, -84, 13],
... [0, -50, 17, 10, 199, -23, -98, 14],
... [0, 15, -5, -2, -59, 7, 30, -2]])
sage: Z, U = C.zigzag_form(transformation=True)
sage: Z
[2|1|0|0|0|0|0|0]
[-+-+-+-+-+-+-+-]
[0|2|0|0|0|0|0|0]
[-+-+-+-+-+-+-+-]
[0|1|2|1|0|0|0|0]
[-+-+-+-+-+-+-+-]
[0|0|0|2|0|0|0|0]
[-+-+-+-+-+-+-+-]
[0|0|0|1|2|1|0|0]
[-+-+-+-+-+-+-+-]
[0|0|0|0|0|2|0|0]
266 Chapter 7. Base class for matrices, part 2