How to determine if a matrix can be decomposed into only shear matrices

Question

My question is that given an arbitrary $2\times2$ matrix with unit determinant, say $$ G_r =\left(\begin{array}{cc}\cos(g d) & -\frac{1}{n_0 g} \sin(g d) \\ n_0 g \sin(g d) & \cos(g d)\end{array}\right)$$ or $$G_r = \begin{pmatrix} \cosh(\hat g d) & -\frac{1}{n_0 \hat g} \sinh(\hat g d) \\ -n_0 \hat g \sinh(\hat g d) & \cosh(\hat g d) \end{pmatrix}$$ where $n_0, g, \hat g, d\in \mathbb{R}$, is there a way to determine if the matrix can be decomposed into the multiplication of a series of shear matrices $R_a = \begin{pmatrix} 1 & 0 \\ a & 1 \end{pmatrix}$ and $T_a = \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix}$ where $a\in \mathbb{R}$, i.e. $G_r = R_{a1}T_{b1}R_{a2}T_{b2}\dots$

My thoughts: It is very clear that those matrices all belong to $SL(2)$ group, which means $det = 1$. I have tried several unsuccessful way of decomposition:

1. LU decomposition: this can decompose the matrix into upper and lower diagonal matrices, but with no guarantee that the diagonal entries are ones.

2. Try to prove ${I, T_a, R_a, G_r}$ forms a subgroup, and $T_a$ and $R_a$ are group generators. I'm not entirely sure about how to prove group generators and hope someone can give me some advice.

Answer 1

Consider the following matrix decompositions into products of shears: \begin{eqnarray} R &=& \begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix} &=& \begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}\begin{bmatrix}1 & 0\\-1 & 1\end{bmatrix}\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix} \\ S_c &=& \begin{bmatrix}c & 0\\0 & c^{-1}\end{bmatrix} &=& \begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}\begin{bmatrix}1 & 0\\c-1 & 1\end{bmatrix}\begin{bmatrix}1 & -c^{-1}\\0 & 1\end{bmatrix}\begin{bmatrix}1 & 0\\c(1-c) & 1\end{bmatrix} \end{eqnarray} Now, consider an arbitrary matrix 2x2 matrix $A$ of unit determinant. If the upper left entry of $A$ is $0$, use $RA$ instead. Find the LDU-decomposition of $A$ (or $RA$). $L$ and $U$ will be shears, since they are triangular matrices with unit diagonal. Since $D$ is a diagonal matrix with unit determinant, it be equal to $S_c$ for some $c$. Thus, either $A = LS_cU$ or $A = RLS_cU$. Since $L$ and $U$ are themselves shears and $R$ and $S_c$ can be decomposed into shears, $A$ can be decomposed into shears.

I think this generalizes to $n\times n$ matrices with unit determinant. All of the row swap-and-negate matrices $R$ can be analogously written as products of shears, the $L$ and $U$ of the LDU-composition are unit triangular matrices which obviously decompose into shears, and a unit determinant diagonal matrix should be factorable into products of $S_c$-like matrices, which themselves decompose into shears.

Follow-up: what's the minimum number of shears needed to represent any unit determinant matrix?