# Thread: Asserting commutability in GL(r,Zn)

1. ## 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...

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!

3. Originally Posted by Ackbeet
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?

4. 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?

5. I want it for a particular case say r=2. Hope it may be generalised to any r.

6. 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.

7. Ok...Same matrices generated with a computer and I want to assert that both A,C are in GL(r,Zn).How to?

8. Non-zero determinant modulo n, and all entries are integers between 0 and n-1 (inclusive)...

9. 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.

10. Originally Posted by Swlabr
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.

11. Originally Posted by jsridhar72
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.

12. Originally Posted by Swlabr
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)?

13. Originally Posted by jsridhar72
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...