Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: a = matrix({(1,1):10, (2,1):-3, (2,2):4/3}); a
[ 0 0 0]
[ 0 10 0]
[ 0 -3 4/3]
sage: a.trace()
34/3
trace_of_product(other)
Returns the trace of self * other without computing the entire product.
EXAMPLES:
sage: M = random_matrix(ZZ, 10, 20)
sage: N = random_matrix(ZZ, 20, 10)
sage: M.trace_of_product(N)
-1629
sage: (M*N).trace()
-1629
visualize_structure(filename=None, maxsize=512)
Write a PNG image to ‘filename’ which visualizes self by putting black pixels in those positions which
have nonzero entries.
White pixels are put at positions with zero entries. If ‘maxsize’ is given, then the maximal dimension in
either x or y direction is set to ‘maxsize’ depending on which is bigger. If the image is scaled, the darkness
of the pixel reflects how many of the represented entries are nonzero. So if e.g. one image pixel actually
represents a 2x2 submatrix, the dot is darker the more of the four values are nonzero.
INPUT:
•filename - either a path or None in which case a filename in the current directory is chosen automatically
(default:None)
maxsize - maximal dimension in either x or y direction of the resulting image. If None or a maxsize
larger than max(self.nrows(),self.ncols()) is given the image will have the same pixelsize as the matrix
dimensions (default: 512)
EXAMPLE:
sage: M = random_matrix(CC, 4)
sage: M.visualize_structure(os.path.join(SAGE_TMP, "matrix.png"))
weak_popov_form(ascend=True)
This function computes a weak Popov form of a matrix over a rational function field k(x), for k a field.
INPUT:
•ascend - if True, rows of output matrix W are sorted so degree (= the maximum of the degrees of the
elements in the row) increases monotonically, and otherwise degrees decrease.
OUTPUT:
A 3-tuple (W, N, d) consisting of:
1.W - a matrix over k(x) giving a weak the Popov form of self
2.N - a matrix over k[x] representing row operations used to transform self to W
3.d - degree of respective columns of W; the degree of a column is the maximum of the degree of its
elements
N is invertible over k(x). These matrices satisfy the relation N ∗ self = W .
EXAMPLES:
262 Chapter 7. Base class for matrices, part 2