Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: a.randomize(0.5)
sage: a
[ 1/2*x^2 - x - 12 1/2*x^2 - 1/95*x - 1/2 0]
[-5/2*x^2 + 2/3*x - 1/4 0 0]
[ -x^2 + 2/3*x 0 0]
Now we randomize all the entries of the resulting matrix:
sage: a.randomize()
sage: a
[ 1/3*x^2 - x + 1 -x^2 + 1 x^2 - x]
[ -1/14*x^2 - x - 1/4 -4*x - 1/5 -1/4*x^2 - 1/2*x + 4]
[ 1/9*x^2 + 5/2*x - 3 -x^2 + 3/2*x + 1 -2/7*x^2 - x - 1/2]
We create the zero matrix over the integers:
sage: a = matrix(ZZ, 2); a
[0 0]
[0 0]
Then we randomize it; the x and y parameters, which determine the size of the random elements, are
passed onto the ZZ random_element method.
sage: a.randomize(x=-2^64, y=2^64)
sage: a
[-12401200298100116246 1709403521783430739]
[ -4417091203680573707 17094769731745295000]
rational_form(format=’right’, subdivide=True)
Returns the rational canonical form, also known as Frobenius form.
INPUT:
•self - a square matrix with entries from an exact field.
•format - default: ‘right’ - one of ‘right’, ‘bottom’, ‘left’, ‘top’ or ‘invariants’. The first four will
cause a matrix to be returned with companion matrices dictated by the keyword. The value ‘invariants’
will cause a list of lists to be returned, where each list contains coefficients of a polynomial associated
with a companion matrix.
•subdivide - default: ‘True’ - if ‘True’ and a matrix is returned, then it contains subdivisions delineating
the companion matrices along the diagonal.
OUTPUT:
The rational form of a matrix is a similar matrix composed of submatrices (“blocks”) placed on the main
diagonal. Each block is a companion matrix. Associated with each companion matrix is a polynomial.
In rational form, the polynomial of one block will divide the polynomial of the next block (and thus, the
polynomials of all subsequent blocks).
Rational form, also known as Frobenius form, is a canonical form. In other words, two matrices are similar
if and only if their rational canonical forms are equal. The algorithm used does not provide the similarity
transformation matrix (also known as the change-of-basis matrix).
Companion matrices may be written in one of four styles, and any such style may
be selected with the format keyword. See the companion matrix constructor,
sage.matrix.constructor.companion_matrix(), for more information about companion
matrices.
If the ‘invariants’ value is used for the format keyword, then the return value is a list of lists, where each
list is the coefficients of the polynomial associated with one of the companion matrices on the diagonal.
224 Chapter 7. Base class for matrices, part 2