Hi, On Tuesday 03 Jul 2012, Simon King wrote: > Hi! > > On 2012-07-02, Martin Albrecht <[email protected]> wrote: > > Shouldn't both give the same distribution mod p? Since every non-singular > > matrix A has a LU decomposition we should be able to just sample L and U > > separately to produce A? > > Sorry for my ignorance, but is it really the case that an LU > decomposition exists for all invertible matrices? I thought there may > only be an LUP decomposition.
Argh, yes, you're right: should be LUP. > If I am not mistaken, the LU decomposition is unique if one requires > that L (or U) has only 1 on the diagonal. Because of the uniqueness, I'd > expect that putting 1 on the diagonal of L and choosing the entries of U > and the remaining of L randomly equally distributed yields a reasonable > distribution of invertible matrices. > > However, if it is really the case that we must consider LUP > decompositions, then I am not totally convinced that a nicely distributed > random choice of a permutation matrix P on top of the choice of L and U as > above yields a nice distribution of invertible matrices. Mhh, why not? If A = LUP we just write AP^-1 = LU, hence for each LU we construct there are as many As as there are permutation matrices, or am I missing something (again :))? Cheers, Martin -- name: Martin Albrecht _pgp: http://pgp.mit.edu:11371/pks/lookup?op=get&search=0x8EF0DC99 _otr: 47F43D1A 5D68C36F 468BAEBA 640E8856 D7951CCF _www: http://martinralbrecht.wordpress.com/ _jab: [email protected] -- To post to this group, send an email to [email protected] To unsubscribe from this group, send an email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
