Let $A \in \mathbb{R}^{m \times n}$ be a matrix with $m \geq n$ of full rank. I'm trying to solve for the integral \begin{equation} f(\vec{x})=\int_{\mathbb{R}^n} \exp(- \lVert A \vec{x} \rVert_1) \, d\vec{x}. \end{equation}
In the case that $m = n$, I can solve the integral by taking the $RQ=A$ factorization of $A$, where $R$ is upper-triangular with positive diagonal and $Q$ is orthonormal: \begin{equation} R = \begin{bmatrix} R_{1,1} & \dots & R_{1,n}\\ & \ddots & \vdots \\ && R_{n,n} \end{bmatrix}, \quad R_{i,i} > 0 \end{equation} Then, via a change of variables $\vec{y}=Q\vec{x}$, we get that \begin{equation} \int_{\mathbb{R}^n} \exp(- \lVert A \vec{x} \rVert_1) \, d\vec{x} = \int_{\mathbb{R}^n} \exp(- \lVert R \vec{y} \rVert_1) \, d\vec{y}. \end{equation}
Then, if $\vec{y} = \begin{bmatrix}y_1 \\ \vdots \\ y_n\end{bmatrix}$, by performing the integrals in order from 1 to $n$, the innermost integral (w.r.t. $y_1$) can be cleanly separated: \begin{align} \int_{-\infty}^\infty \exp\left(-\left\lvert \sum_{i=1}^n R_{1,i} \, y_i \right\rvert\right) \, dy_1 &= \int_{-\infty}^{c_1} \exp\left(\sum_{i=1}^n R_{1,i} \, y_i \right) \, dy_1 + \int_{c_1}^{\infty} \exp\left(-\sum_{i=1}^n R_{1,i} \, y_i \right) \, dy_1 \\ &= \frac{2}{R_{11}}. \end{align} Repeating this process for $y_2, \dots, y_n$ then yields the answer \begin{equation} f(\vec{x}) = \frac{2^n}{ \prod_{i=1}^n R_{ii}} \end{equation}
Is there any way to generalize this to the case for $m>n$?
One initial thought was that the QR transform seems to be applicable to rectangular matrices as well, but I couldn't find any information about whether RQ exists as well. If the RQ transform did exist, perhaps I could do the same thing but with $R \in \mathbb{R}^{m \times n}, Q \in \mathbb{R}^{n \times n}$ instead?