I sent the following to John Cremona - he suggested I post it here.
Dear John -
I was computing Smith normal forms (over ZZ) and I ran into an issue. I
boiled it down to this:
sage: version()
'Sage Version 5.12, Release Date: 2013-10-07'
sage: M = matrix([[1,0],[0,1]])
sage: M
[1 0]
[0 1]
sage: M.parent()
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
sage: M.inverse()
[1 0]
[0 1]
sage: M.inverse().parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
Note that the base ring has changed. This caught me out: I was computing
Smith normal forms in two ways and they were different. Eventually I
realized that (obviously) the Smith normal form is sensitive to the base
ring!
Does the base ring have to change here? I guess that there is a subtle
algorithm to quickly compute inverses, that requires extending the ring.
But once we've got the answer, couldn't we do a fast check to see if the
answer is actually in the original base ring?
That is - if M is the original matrix and N is the computed inverse then
add this to the very end:
if N.parent() == M.parent():
return N
try:
R = M.parent().base_ring()
P = N.change_ring(R)
assert M*P == P*M == 1
return P
except:
return N
all the best,
saul
[Edit]
John pointed me to the code for M.__invert__() so I suppose that I am
suggesting placing some version of the above code after the try and before
the return, around line 11.
"try:
return (~self.lift()).change_ring(self.base_ring())"
Of course, in this particular place in the code the first two lines of my
suggestion are pointless.
best,
saul
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