> M=random_matrix(IntegerModRing(10),3,3)
>
> Now the adjoint certainly exists, but M.adjoint() isn't as yet
> implemented over general rings - only over ZZ or QQ. How can I use,
> say, Maxima or Pari within Sage to compute the adjoint of M?
>
It's pretty friendly to use Pari to do this ... in general, given any
sage object a, you can try pari(a) to convert it to Pari.
In this example, it works fine (though the pari output isn't the most
pleasant to read):
sage: M = random_matrix(IntegerModRing(10),3,3)
sage: Mp = pari(M)
sage: M
[4 2 0]
[5 4 8]
[9 6 2]
sage: Mp
[Mod(4, 10), Mod(2, 10), Mod(0, 10); Mod(5, 10), Mod(4, 10), Mod(8,
10); Mod(9, 10), Mod(6, 10), Mod(2, 10)]
Then you can call Mp.matadjoint() to get the adjoint. However, in this
case, Pari fails to find the adjoint:
sage: Mp.matadjoint()
---------------------------------------------------------------------------
PariError Traceback (most recent call last)
/Users/craigcitro/.sage/temp/sharma.local/8072/_Users_craigcitro__sage_init_sage_0.py
in <module>()
----> 1
2
3
4
5
/sage/local/lib/python2.5/site-packages/sage/libs/pari/gen.so in
sage.libs.pari.gen._pari_trap (sage/libs/pari/gen.c:38577)()
8048
8049
-> 8050
8051
8052
PariError: impossible inverse modulo: (36)
This is actually Pari behaving badly, not sage -- doing the same thing
in a GP session gets the same error. Apparently Pari can't find the
determinant of non-invertible square matrices of size at least 3x3
(???). I'll try to look at this and submit a bug report soon -- or you
could beat me to it. :)
In any event, say it worked (so I'll use 2x2 as an example). You
*should* be able to just call M.parent()(Mp.matadjoint()), but that
currently hits an error. However, the pari object Mp.matadjoint() has
a _sage_ method that knows how to convert back to sage. So this works:
sage: m = random_matrix(Integers(10),2,2) ; m
[0 9]
[3 5]
sage: mp = pari(m)
sage: mp.matadjoint()
[Mod(5, 10), Mod(1, 10); Mod(7, 10), Mod(0, 10)]
sage: mp.matadjoint()._sage_()
[5 1]
[7 0]
sage: pari(m).matadjoint()._sage_()
[5 1]
[7 0]
As you point out at the beginning, though, we really should have our
own native adjoint function. Here's a one-liner:
sage: M = random_matrix(Integers(10),3,3) ; M
[9 7 5]
[2 9 5]
[7 8 8]
sage: M.parent()([ (-1)^(i+j)*M.matrix_from_rows_and_columns([x for x
in range(M.nrows()) if x != i], [y for y in range(M.ncols()) if y !=
j]).det() for i in range(M.nrows()) for j in range(M.nrows())
]).transpose()
[2 4 0]
[9 7 5]
[3 7 7]
That function is terrible in a lot of ways -- but if you need
something right now, it works. :)
-cc
--~--~---------~--~----~------------~-------~--~----~
To post to this group, send email to sage-support@googlegroups.com
To unsubscribe from this group, send email to
sage-support-unsubscr...@googlegroups.com
For more options, visit this group at
http://groups.google.com/group/sage-support
URLs: http://www.sagemath.org
-~----------~----~----~----~------~----~------~--~---
