Hello,
I hope everyone is fine. I have a small question.
Suppose M denotes the set of all 2x2 matrices with integer entries. I know how to randomly pick an element from this space (by using the sage command .random_element().
My question is:
How can I uniformly pick a matrix from this space. I am not able to find any command for that? Kindly help.
Let's create the space M:
sage: M = MatrixSpace(ZZ,2,2)
sage: M
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
The default behavior is:
sage: M.random_element()
[-2 1]
[-3 0]
One may give an interval for the values:
sage: M.random_element(x=1000, y=2000)
[1308 1271]
[1451 1352]
One may increase the density of zero entries in the elements:
sage: M.random_element(x=1000, y=2000, density=.9)
[ 0 1366]
[ 0 1275]
One may provide a distribution, for instance the gaussian distribution with parameter sigma=x centered at zero. Below, I translate the center at 10^9:
sage: M.random_element(distribution='gaussian', x=10^9) + 10^9 * M((1,1,1,1))
[-203915342 -852236679]
[ 222400125 1463169355]
For further information and other possible inputs, read the below documentation :
sage: M.random_element?
sage: z = M.zero_matrix()
sage: z.randomize?
sage: ZZ.random_element?
As @slelievre said, there is no such thing as uniform distribution for integer matrices, which form an infinite countable set. If the probability of every such matrix is the same, since there are infinitely such matrices, and the total probability is 1, the common probability of every single matrix should be zero. But since there are countably integer matrices, then the total probability is 0 (because the measure is sigma-additive), not 1.
Asked: 2022-08-24 12:59:53 +0200
Seen: 226 times
Last updated: Aug 25 '22
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From a countable set, what would it mean to pick uniformly at random?