Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
•subdivide - default: True - if True the ZigZag form matrix is subdivided according to the
companion matrices described in the output section below.
OUTPUT:
A matrix in ZigZag form has blocks on the main diagonal that are companion matrices. The first companion
matrix has ones just below the main diagonal. The last column has the negatives of coefficients of a
monic polynomial, but not the leading one. Low degree monomials have their coefficients in the earlier
rows. The second companion matrix is like the first only transposed. The third is like the first. The fourth
is like the second. And so on.
These blocks on the main diagonal define blocks just off the diagonal. To the right of the first companion
matrix, and above the second companion matrix is a block that is totally zero, except the entry of the first
row and first column may be a one. Below the second block and to the left of the third block is a block
that is totally zero, except the entry of the first row and first column may be one. This alternating pattern
continues. It may now be apparent how this form gets its name. Any other entry of the matrix is zero. So
this form is reminiscent of rational canonical form and is a good precursor to that form.
If transformation is True, then the output is a pair of matrices. The first is the form Z and the second
is an invertible matrix U such that U.inverse()*self*U equals Z. In other words, the repsentation of
self with respect to the columns of U will be Z.
If subdivide is True then the matrix returned as the form is partitioned according to the companion
matrices and these may be manipulated by several different matrix methods.
For output that may be more useful as input to other routines, see the helper method _zigzag_form().
Note: An efffort has been made to optimize computation of the form, but no such work has been done
for the computation of the transformation matrix, so for fastest results do not request the transformation
matrix.
ALGORITHM:
ZigZag form, and its computation, are due to Arne Storjohann and are described in [STORJOHANN-
THESIS] and [STORJOHANN-ISACC98], where the former is more representative of the code here.
EXAMPLES:
Two examples that illustrate ZigZag form well. Notice that this is not a canonical form. The two matrices
below are similar, since they have equal Jordan canonical forms, yet their ZigZag forms are quite different.
In other words, while the computation of the form is deterministic, the final result, when viewed as a
property of a linear transformation, is dependent on the basis used for the matrix representation.
sage: A = matrix(QQ, [[-68, 69, -27, -11, -65, 9, -181, -32],
... [-52, 52, -27, -8, -52, -16, -133, -14],
... [ 92, -97, 47, 14, 90, 32, 241, 18],
... [139, -144, 60, 18, 148, -10, 362, 77],
... [ 40, -41, 12, 6, 45, -24, 105, 42],
... [-46, 48, -20, -7, -47, 0, -122, -22],
... [-26, 27, -13, -4, -29, -6, -66, -14],
... [-33, 34, -13, -5, -35, 7, -87, -23]])
sage: Z, U = A.zigzag_form(transformation=True)
sage: Z
[ 0 0 0 40| 1 0| 0 0]
[ 1 0 0 52| 0 0| 0 0]
[ 0 1 0 18| 0 0| 0 0]
[ 0 0 1 -1| 0 0| 0 0]
[---------------+-------+-------]
[ 0 0 0 0| 0 1| 0 0]
265