Results 1 to 13 of 13

Math Help - Asserting commutability in GL(r,Zn)

  1. #1
    Junior Member
    Joined
    Mar 2010
    Posts
    26

    Asserting commutability in GL(r,Zn)

    I have two matrices A and C from GL(r,Zn).How can I assert that they wont commute.That is how can I ensure the AC<>CA(AC not equals CA). thanks in advance...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    5
    Awards
    2
    You say you "have" two matrices. Does that mean you have the exact matrix representation of these two matrices? If so, why not just compute AC and CA, and see if they're different? Some matrices in your group will commute, and others will not. The identity, for example, commutes with all members of the group!
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Mar 2010
    Posts
    26
    Quote Originally Posted by Ackbeet View Post
    You say you "have" two matrices. Does that mean you have the exact matrix representation of these two matrices? If so, why not just compute AC and CA, and see if they're different? Some matrices in your group will commute, and others will not. The identity, for example, commutes with all members of the group!
    I randomly generate the matrices say using a computer. The computer outputs yes if commutable and no otherwise. Get it?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    5
    Awards
    2
    So you want a condition on the matrices that will tell you in advance, without computing AC and CA, whether the two products will be equal. Is that correct?

    I have a question: what size matrices are you dealing with here? That is, what is r?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Mar 2010
    Posts
    26
    I want it for a particular case say r=2. Hope it may be generalised to any r.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    5
    Awards
    2
    Well, you could try doing this: let

    A=\begin{bmatrix}a_{1} &a_{2}\\ a_{3} &a_{4}\end{bmatrix} and

    C=\begin{bmatrix}c_{1} &c_{2}\\ c_{3} &c_{4}\end{bmatrix}.

    Then

    AC=\begin{bmatrix}a_{1}c_{1}+a_{2}c_{3} &a_{1}c_{2}+a_{2}c_{4}\\ a_{3}c_{1}+a_{4}c_{3} &a_{3}c_{2}+a_{4}c_{4}\end{bmatrix}.

    Similarly,

    CA=\begin{bmatrix}a_{1}c_{1}+a_{3}c_{2} &a_{2}c_{1}+a_{4}c_{2}\\ a_{1}c_{3}+a_{3}c_{4} &a_{2}c_{3}+a_{4}c_{4}\end{bmatrix}.

    Therefore, the commutator [A,C]\equiv AC-CA we compute as follows:

    [A,C]=\begin{bmatrix}a_{2}c_{3}-a_{3}c_{2} &a_{1}c_{2}+a_{2}c_{4}-a_{2}c_{1}-a_{4}c_{2}\\ a_{3}c_{1}+a_{4}c_{3}-a_{1}c_{3}-a_{3}c_{4} &a_{3}c_{2}-a_{2}c_{3}\end{bmatrix}.

    For the two matrices to commute, the commutator must be equal to zero. Interesting point: given either A or C, you can view the equation [A,C]=0 as a system of four linear equations in four unknowns for the unknown matrix. You can then characterize, to some extent, all the matrices that will commute with a given matrix. Note: since the zero matrix and the identity matrix both commute with all matrices, there will always be infinitely many matrices that commute with a given matrix (although, of course, since you're in Z_{n}, that will change things a bit. You might have a finite number of commuting matrices in this case.)

    That's about as far as I can go.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Mar 2010
    Posts
    26
    Ok...Same matrices generated with a computer and I want to assert that both A,C are in GL(r,Zn).How to?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    Non-zero determinant modulo n, and all entries are integers between 0 and n-1 (inclusive)...
    Follow Math Help Forum on Facebook and Google+

  9. #9
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    5
    Awards
    2
    So, based on Swlabr's comment, that'll give you two more equations (nonlinear this time, unfortunately) to help characterize your matrices. So you're up to six equations (2 of which are nonlinear, 4 linear) in 8 unknowns.

    [EDIT] Actually, I'm wrong. Determinant nonzero gives you two nonlinear inequalities. That's just a check condition at the end, they won't help you solve the system.
    Last edited by Ackbeet; August 18th 2010 at 05:40 AM. Reason: Inequalities.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Junior Member
    Joined
    Mar 2010
    Posts
    26
    Quote Originally Posted by Swlabr View Post
    Non-zero determinant modulo n, and all entries are integers between 0 and n-1 (inclusive)...
    Sorry for late reply. MHF was not available to me for the last couple of days.

    Anyway Since all matrices are invertible in GL(r,Zn) the determinant is not zero. Then your statement is bit confusing or I misunderstood? Please clarify.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    Quote Originally Posted by jsridhar72 View Post
    Sorry for late reply. MHF was not available to me for the last couple of days.

    Anyway Since all matrices are invertible in GL(r,Zn) the determinant is not zero. Then your statement is bit confusing or I misunderstood? Please clarify.
    Well, you were wanting to find a way of knowing if two matrices are in GL(r, Z_n), and this is the most obvious way of doing it.
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Junior Member
    Joined
    Mar 2010
    Posts
    26
    Quote Originally Posted by Swlabr View Post
    Well, you were wanting to find a way of knowing if two matrices are in GL(r, Z_n), and this is the most obvious way of doing it.
    Yes Sir, I agree. Is there any connection for commutability of matrices in GL(r,Zn)and inverse in GL(r,Zn)?
    Follow Math Help Forum on Facebook and Google+

  13. #13
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    Quote Originally Posted by jsridhar72 View Post
    Yes Sir, I agree. Is there any connection for commutability of matrices in GL(r,Zn)and inverse in GL(r,Zn)?
    I doubt it. One can prove that S_n \cong GL(n, 1), and so every finite group can be embedded into such a group, and so basically your problem is `reduced' to finding the center of an arbitrary group...
    Follow Math Help Forum on Facebook and Google+

Search Tags


/mathhelpforum @mathhelpforum