Symbolic matrices and "integrity" of their inverse

asked 2011-10-17 12:44:30 +0200

270 ●1 ●7 ●19

updated 2011-10-18 05:46:11 +0200

I have to solve the following problem:

Does a matrix $G\in GL(n,\mathbb{Z})$ exists such that $$ G\times A\times G^{-1}=B $$ being $A,B$ given matrices in $\mathbb{Q}$?

Doing everything by hand, I finally find myself with a bunch of symbolic matrices. Now I have to check if they can lay inside $GL(n,\mathbb{Z})$, i.e. if there are integer values for the variables in the matrix such that the matrix is integer, invertible and with integer inverse.

E.g.: $$\left(\begin{array}{cc}x & 0 \\ 0 & y\end{array}\right)$$ does the trick only for $x=y=1$.

Is there a quick method within Sage to solve that last problem?

Thanks!

edit retag flag offensive close merge delete

Comments

An integer matrix is in GL(n,Z) (i.e., invertible with integer inverse) if and only if its determinant is 1 or -1 (from wikipedia: a square matrix over a commutative ring is invertible if and only if its determinant is a unit in that ring.) This might help you...

parzan ( 2011-10-18 08:28:07 +0200 )edit

It does help me indeed, thanks! I should have known that... My main concern now is how to find out whether a polynomial equation (det=+/-1) has integer roots or not. "assume()" does not the trick :(

Jesustc ( 2011-10-18 11:51:33 +0200 )edit

This is in general an undecidable question - see http://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem. However, for specific cases you might be able to find a solution.

parzan ( 2011-10-18 12:08:40 +0200 )edit

"Integrality" would be a better word than "integrity".

John Palmieri ( 2011-10-18 13:55:10 +0200 )edit

Thanks a lot for the comments, Parzan and Palmieri! What do you think about erasing the question? I don't think it is useful for anybody.

Jesustc ( 2011-10-28 11:22:31 +0200 )edit

add a comment

answered 2013-06-26 18:14:35 +0200

tmonteil
27323 ●31 ●202 ●514 http://wiki.sagemath.o...

A direct approach (not using symbolic solving) could be:

sage: A = matrix(QQ,[[3,1],[2,4]]) ; A
[3 1]
[2 4]
sage: B = matrix(QQ,[[2,4],[0,5]]) ; B
[2 4]
[0 5]
sage: A.is_similar(B)                       
True
sage: T = A.is_similar(B,transformation=True)[1] ; T
[ 1.000000000000000?             0.?e-18]
[0.2500000000000000? 0.2500000000000000?]
sage: T.change_ring(QQ)
[  1   0]
[1/4 1/4]
sage: G = T.change_ring(QQ)/det(T) ; G
[4 0]
[1 1]
sage: G * A * G^(-1) == B
True

See A.is_similar? for details.

edit flag offensive delete link

Comments

That's a good one, thanks!

Jesustc ( 2013-08-03 12:26:35 +0200 )edit

add a comment

Symbolic matrices and "integrity" of their inverse

Comments

1 Answer

Comments

Your Answer

Question Tools

Stats

Related questions

Symbolic matrices and "integrity" of their inverse edit

Comments

1 Answer

Comments

Your Answer

Question Tools

Stats

Related questions

Symbolic matrices and "integrity" of their inverse