Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: R = PolynomialRing(QQ, ’x12,x13,x14,x23,x24,x34’)
sage: x12, x13, x14, x23, x24, x34 = R.gens()
sage: A = matrix(R, [[ 0, x12, x13, x14],
....: [-x12, 0, x23, x24],
....: [-x13, -x23, 0, x34],
....: [-x14, -x24, -x34, 0]])
sage: A.pfaffian()
x14*x23 - x13*x24 + x12*x34
sage: parent(A.pfaffian())
Multivariate Polynomial Ring in x12, x13, x14, x23, x24, x34 over Rational Field
The Pfaffian of an alternating matrix squares to its determinant:
sage: A = [[0] * 6 for i in range(6)]
sage: for i in range(6):
....: for j in range(i):
....: u = floor(random() * 10)
....: A[i][j] = u
....: A[j][i] = -u
....: A[i][i] = 0
sage: AA = Matrix(ZZ, A)
sage: AA.pfaffian() ** 2 == AA.det()
True
AUTHORS:
•Darij Grinberg (1 Oct 2013): first (slow) implementation.
pivot_rows()
Return the pivot row positions for this matrix, which are a topmost subset of the rows that span the row
space and are linearly independent.
OUTPUT: a tuple of integers
EXAMPLES:
sage: A = matrix(QQ,3,3, [0,0,0,1,2,3,2,4,6]); A
[0 0 0]
[1 2 3]
[2 4 6]
sage: A.pivot_rows()
(1,)
sage: A.pivot_rows() # testing cached value
(1,)
plot(*args, **kwds)
A plot of this matrix.
Each (ith, jth) matrix element is given a different color value depending on its relative size compared to
the other elements in the matrix.
The tick marks drawn on the frame axes denote the (ith, jth) element of the matrix.
This method just calls matrix_plot. *args and **kwds are passed to matrix_plot.
EXAMPLES:
A matrix over ZZ colored with different grey levels:
sage: A = matrix([[1,3,5,1],[2,4,5,6],[1,3,5,7]])
sage: A.plot()
222 Chapter 7. Base class for matrices, part 2