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.
2 Answers
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1
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 ]
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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
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]
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