It's well known that $2 \times 2$ complex matrices act on the complex plane (with a point at infinity) by $\begin{pmatrix}a & b \\ c & d \end{pmatrix}\cdot z = \frac{az + b}{cz + d}$.
Based on this, various matrix decompositions for $2 \times 2$ matrices gain simple meanings in planar geometry. Does the QR decomposition gain such a meaning? And if so, what is it?
In general the QR decomposition breaks a linear map down into an orthogonal (or unitary in the complex case) map and one that preserves a fixed flag. A flag here is a chain of subspaces $V_1 \leq V_2 \leq \cdots \leq V_n = \mathbb{C}^n$. In this example, a flag is exactly a line $L \leq \mathbb{C}^2$
We can think of the unitary matrices here as the ones which preserve a metric on the projective space and the upper triangular matrices as those which preserve a fixed line. So every transformation is the product of two transformations, one of which preserves the matric and one of which preserves a line.
Note immediately that this means $SU(2) \leq SL(2,\mathbb{C})$ acts transitively on the complex projective line.
More generally, we can see this fitting into the broader geometric picture that, for a real Lie group $G$ with a Borel subgroup $B$ (or any parabolic subgroup in fact), a maximal compact subgroup $K$ acts transitively on $G/B$. Alternatively, we could say this as the $G$-orbits of parabolic subgroups of $G$ are the same as the $K$-orbits.