I know the two generators of $SL_2(\mathbb{Z})$ are $S=\begin{bmatrix} 0 & -1 \\ 1& 0 \end{bmatrix}$ and $T=\begin{bmatrix} 1 & 1 \\ 0& 1 \end{bmatrix}$.
Furthermore, $T^n =\begin{bmatrix} 1 & n \\ 0& 1 \end{bmatrix}$.
Suppose I have any matrix $A$ in $SL_2(\mathbb{Z})$. I want to find its generators. Here is what I have done so far:
I am trying to find the $a,b,c,d \in \mathbb{N}$ such that
$A=S^a T^b S^{-c} T^{-d} \iff A T^d S^c= S^a T^b$. Also, since $S^4=1$, I assumed $1 \leq a,c \leq 4$.
In general, I tried to compare the coefficients of $A$ for every possible values of $a$ and $c$, without success.
Where is my mistake?
Following Diamond's book p.21, the idea is to right multiply by $T^{-n}S$ to replace the bottom row $(c,d)$ by $(d-nc,-c),n = \lfloor d/c \rfloor$ ie. one iteration of the $\gcd$ algorithm, to reduce iteratively the bottom left coefficient $c$ until it is $0$.
Once this is done then $ad-bc = 1$ implies the matrix is $T^b$.
Thus the obtained decomposition is $A = (-1)^jT^b\prod_{i=j}^1 (S T^{n_i})$