Find the invariant conic of an elliptic element of $SL_2(\mathbb R)$

Question

Let $M$ be an elliptic element of $SL_2(\mathbb R)$. Then it is conjugate to a rotation $R(\theta)$. Note that we can calculate $\theta$ in terms of the trace of $M$; it means that we actually know $R(\theta)$ and we can write:

$$M=TR(\theta) T^{-1}$$

If $S^1$ is the unit circle in $\mathbb R^2$, it follows that $T(S^1)$ is the conic section $\mathcal C$ which is preserved by $M$.

Is there any explicit way to find the equation $\mathcal C$ in general?

My procedure is quite uneffective, because one has to find $T$ first (so non-linear system) and then write down $T(S^1)$, which is in general not obvious.

Answer 1

Here's an approach you might like: let $\lambda = \cos \theta + i \sin \theta$. Let $v \in \Bbb C^2$ denote the eigenvector of $M$ associated with $\lambda$. Then the matrix $T$ whose columns are the real and complex parts of $v$ are such that $$ M = TR(-\theta)T^{-1}. $$ So, this $T$ is such that $T(S^1)$ is the conic section preserved by $M$.

To see that this is the case, note that $$ Mv = \lambda v = (\cos \theta + i \sin \theta) (v_R + iv_I) = (\cos \theta v_R - \sin \theta v_I) + i(\sin \theta v_R + \cos \theta v_I). $$ It follows that $$ Mv_R = \operatorname{Re}[Mv] = \cos \theta \,v_R - \sin \theta v_I,\\ Mv_I = \operatorname{Im}[Mv] = \sin \theta v_R + \cos \theta v_I. $$ Once you have $T$, it is not too difficult to find $T(S^1)$.

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We can make things even more direct. Note that $$ M^\top = T^{-\top}R(\theta) T^\top. $$ Find the eigenvector $v$ of $M^\top$ associated with $\lambda$. The real and imaginary parts of this eigenvector form the rows of $T^{-1}$. Now, the equation of $T(S^1)$ is given by $\|T^{-1}(x_1,x_2)\| = 1$. In other words, $$ (v_{R,1} x_1 + v_{R,2}x_2)^2 + (v_{I,1}x_1 + v_{I,2} x_2)^2 = 1. $$ Note that $v$ can be explicitly expressed in terms of the entries of $M$ as $$ v = (m_{21}, \lambda - m_{11}). $$

Answer 2

While the question has been already answered a while ago more than satisfactorily by Ben, I wanted to offer another perspective; something that I came upon in my recent work (note: not a mathematician). It seems to be a general feature of symplectic groups and since $SL(2,\mathbb{R}) = Sp(2,\mathbb{R})$, it applies also here.

Let $M$ be an element of $Sp(2n,\mathbb{F})$, where $\mathbb{F}$ is a field. This means that $M$ preserves the symplectic structure: $$ M^T\Omega M = \Omega,\quad\textrm{where}\quad \Omega = \begin{pmatrix} 0 && I_n \\ -I_n && 0 \end{pmatrix}. $$ It is easy to show that $M$ also preserves the quadratic form $v^T Q v$ given by matrix $Q = \Omega M$, since $$ (Mv)^T Q (Mv) = v^T M^T \Omega M M v = v^T \Omega M v = v^T Q v. $$ In fact, any linear combination $\sum c_{km} (M^T)^k\Omega M^m$ will do. We can choose the quadratic form to be symmetric, for example $$ Q_s = \frac{1}{2}(Q + Q^T) = \frac{1}{2} (\Omega M - M^T\Omega). $$ With some initially chosen vector $v_0$, the equation $$ v^T Q_s v - v_0^T Q_s v_0 = 0 $$ explicitly defines a quadric surface in $\mathbb{F}^{2n}$. I believe that this quadric is then the orbit of $M$ passing through $v_0$, as any point on the quadric will be mapped by $M$ onto another point on the same surface.

All right, let's see how this looks like in $SL(2,\mathbb{R})$. With $$ M = \begin{pmatrix}a && b \\ c && d\end{pmatrix} $$ we find that $$ Q_s = \begin{pmatrix} c && \frac{d-a}{2} \\ \frac{d-a}{2} && -b \end{pmatrix}. $$ Then we choose an initial vector $v_0 = (x_0\ y_0)^T$ and denote $v_0^T Q_s v_0 = e$. The invariant conic through $v_0$ is then defined by the equation $$ cx^2 + (d-a)xy - by^2 - e = 0. $$ Such conics are classified according to the determinant of $Q$, which is (using $ad-bc=1$) $$ \Delta = -bc - \left(\frac{d-a}{2}\right)^2 = 1 - \left(\frac{a+d}{2}\right)^2 = 1 - \left(\frac{\textrm{Tr}M}{2}\right)^2. $$ For example, we have an ellipse for $\Delta > 0 \Rightarrow \lvert\textrm{Tr}M\rvert < 2$, so the classification of these conic sections parallels the classification of elements of $SL(2,\mathbb{R})$.

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I wonder if what I sketched out here is a well-known fact in mathematics. I really struggled to find any references. The closest thing I saw is Peter J. Olver's Classical Invariant Theory, which discusses orbits of transformation groups, but the explicit form of $Q$ or $Q_s$ never appears. Another one is Geometry of Möbius Transformations by Vladimir V. Kisil, which specifically treats $SL(2,\mathbb{R})$, but he is mostly concerned with hyperbolic geometry, which is beyond my capacity :) I would appreciate a comment from someone better versed in mathematics.