2
$\begingroup$

I need to compute a group relator for a group presentation. In particular, $$A=\begin{pmatrix} a & 0 \\ 0 & 1/a \end{pmatrix}, B=\begin{pmatrix} x & y \\ z & w \end{pmatrix}, $$ where $a$ is nonzero and the determinant of $B$ is 1. I need to expand this matrix mutiplication $$A^{-2}BAB^{-3}ABA^{-1}B^{-2}$$ in terms of coordinates $(a, x,y,z,w)$, I have tried MatrixFunction but it seems only works with one matrix as a variable, and here I have two. Could you help? Thank you.

$\endgroup$
2
  • 3
    $\begingroup$ Try: ma = {{a, 0}, {0, 1/a}}; mb = {{x, y}, {z, w}}; MatrixPower[ma, -2] . mb . ma . MatrixPower[mb, -3] . MatrixPower[ma, -1] . MatrixPower[mb, -2] $\endgroup$ Commented Jul 3 at 11:19
  • $\begingroup$ Hi, can you submit this as an answer so I can accept your answer? Doing it term by term instead of a functions work for me. Also can you tell me how to incorporate the condition wx-yz=1 and simplify the answer? $\endgroup$
    – Han
    Commented Jul 3 at 12:28

2 Answers 2

4
$\begingroup$

You can use MatrixPower for exponentiation as:

ma = {{a, 0}, {0, 1/a}}; 
mb = {{x, y}, {z, w}}; 
res= MatrixPower[ma, -2] . mb . ma . MatrixPower[mb, -3] .   MatrixPower[ma, -1] . MatrixPower[mb, -2]

To simplify and use w x-y z=1 you may write:

Simplify[res, w  x - y  z == 1 ]

enter image description here

$\endgroup$
1
  • $\begingroup$ Thank you, this is very helpful. $\endgroup$
    – Han
    Commented Jul 4 at 9:01
0
$\begingroup$

I suggest that you expand the matrix multiplication $(A^{-2}BAB^{-3}ABA^{-1}B^{-2})$ in terms of the coordinates $((a, x, y, z, w))$ using the following code. This code defines the matrices $A$ and $B$, computes the inverses, and performs the matrix multiplications step-by-step. Here is an attempt to do this which may help you:

(* Define matrices A and B *)
A = {{a, 0}, {0, 1/a}};
B = {{x, y}, {z, w}};

(* Compute inverses of A and B *)
Ainv = Inverse[A];
Binv = Inverse[B];

(* Define the expression *)
expr = Ainv.Ainv.B.A.Binv.Binv.Binv.A.B.Ainv.Binv.Binv;

(* Simplify the expression *)
SimplifiedExpr = Simplify[expr]

(* Output the result *)
SimplifiedExpr

enter image description here

enter image description here

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.