Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
First, note that the density parameter does not ensure the density of a matrix, it is only an upper bound.
sage: A = random_matrix(GF(127),200,200,density=0.3)
sage: A.density()
5211/20000
sage: A = matrix(QQ,3,3,[0,1,2,3,0,0,6,7,8])
sage: A.density()
2/3
sage: a = matrix([[],[],[],[]])
sage: a.density()
0
derivative(*args)
Derivative with respect to variables supplied in args.
Multiple variables and iteration counts may be supplied; see documentation for the global derivative()
function for more details.
EXAMPLES:
sage: v = vector([1,x,x^2])
sage: v.derivative(x)
(0, 1, 2*x)
sage: type(v.derivative(x)) == type(v)
True
sage: v = vector([1,x,x^2], sparse=True)
sage: v.derivative(x)
(0, 1, 2*x)
sage: type(v.derivative(x)) == type(v)
True
sage: v.derivative(x,x)
(0, 0, 2)
det(*args, **kwds)
Synonym for self.determinant(...).
EXAMPLES:
sage: A = MatrixSpace(Integers(8),3)([1,7,3, 1,1,1, 3,4,5])
sage: A.det()
6
determinant(algorithm=None)
Returns the determinant of self.
ALGORITHM:
For small matrices (n less than 4), this is computed using the naive formula. In the specific case of matrices
over the integers modulo a non-prime, the determinant of a lift is computed over the integers. In general, the
characteristic polynomial is computed either using the Hessenberg form (specified by "hessenberg")
or the generic division-free algorithm (specified by "df"). When the base ring is an exact field, the
default choice is "hessenberg", otherwise it is "df". Note that for matrices over most rings, more
sophisticated algorithms can be used. (Type A.determinant to see what is done for a specific matrix
A.)
INPUT:
•algorithm - string:
– "df" - Generic O(n^4) division-free algorithm
150 Chapter 7. Base class for matrices, part 2