Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
Note: The rank is cached.
ALGORITHM: First check if the matrix has maxim possible rank by working modulo one random prime.
If not call LinBox’s rank function.
EXAMPLES:
sage: a = matrix(ZZ,2,3,[1..6]); a
[1 2 3]
[4 5 6]
sage: a.rank()
2
sage: a = matrix(ZZ,3,3,[1..9]); a
[1 2 3]
[4 5 6]
[7 8 9]
sage: a.rank()
2
Here’s a bigger example - the rank is of course still 2:
sage: a = matrix(ZZ,100,[1..100^2]); a.rank()
2
rational_reconstruction(N)
Use rational reconstruction to lift self to a matrix over the rational numbers (if possible), where we view
self as a matrix modulo N.
INPUT:
•N - an integer
OUTPUT:
•matrix - over QQ or raise a ValueError
EXAMPLES: We create a random 4x4 matrix over ZZ.
sage: A = matrix(ZZ, 4, [4, -4, 7, 1, -1, 1, -1, -12, -1, -1, 1, -1, -3, 1, 5, -1])
There isn’t a unique rational reconstruction of it:
sage: A.rational_reconstruction(11)
Traceback (most recent call last):
...
ValueError: Rational reconstruction of 4 (mod 11) does not exist.
We throw in a denominator and reduce the matrix modulo 389 - it does rationally reconstruct.
sage: B = (A/3 % 389).change_ring(ZZ)
sage: B.rational_reconstruction(389) == A/3
True
TEST:
Check that ticket #9345 is fixed:
sage: A = random_matrix(ZZ, 3, 3)
sage: A.rational_reconstruction(0)
Traceback (most recent call last):
332 Chapter 17. Dense matrices over the integer ring