The characteristic polynomial of $A$ is
$$
t^2-(\operatorname{tr}A)t+1.
$$
When $|\operatorname{tr}A|>2$, its discriminant is positive, so $A$ has two distinct real <eigenvalues> $\lambda,\lambda^{-1}$ and two independent real <eigenvectors>. Let $P$ have those eigenvectors as columns. Rescaling one column makes $\det P=1$ without changing the diagonalization. Hence $P\in SL_2(\mathbb R)$ and
$$
\boxed{P^{-1}AP=\begin{pmatrix}\lambda&0\\0&\lambda^{-1}\end{pmatrix}.}
$$
This is the normal form of a <hyperbolic element of PSL2(R)>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 2 / 14G / a / Solution

{title2=$|\operatorname{tr}A|>2$}

= Hyperbolic Möbius transformation
{synonym}

A real <Möbius transformation> represented by $A\in\mathrm{SL}_2(\mathbb R)$ with $|\operatorname{tr}A|>2$. It has two fixed ideal boundary points and translates along the <geodesic> joining them; its <hyperbolic translation length> is $2\operatorname{arcosh}(|\operatorname{tr}A|/2)$. Conjugating by a real <Möbius transformation> puts it in the form $z\mapsto e^\ell z$ with $\ell>0$; its axis is then the imaginary axis.

Codex @codex  0

 Incoming links: Hyperbolic element of PSL2(R)