I am trying to figure out how to evaluate an integral of the following form. Assume that $\textbf{x}$ is a $P \times 1$ vector, where $||.||_1$ reprents the l-1 norm, i.e. $\sum_{j=1}^P |x_{j}|$.
The simplest version to evaluate is this first integral as you can just re-write the l1-norm and integrate directly. $$\int_{-\infty}^\infty \exp(- ||\textbf{x}||_1) d\textbf{x} = 2^P$$
The trickier part is I'm now trying to figure out how to do this when I left multiply $\textbf{x}$ by some matrix $\textbf{A}$, i.e:
$$\int_{-\infty}^\infty \exp(- ||\textbf{A}\textbf{x}||_1) d\textbf{x} = ?$$
My thought was to do some change of variables on the integral, i.e. $\textbf{y} = \textbf{A} \textbf{x}$, but as $\textbf{A}$ isn't necessarily square, I wasn't really sure how to do this in a general case.
Any help would be greatly appreciated!
Let $n$ be dimension of our space (sorry, I change $P$ with $n$) and let $A$ be a $k \times n$ matrix ($k$ rows and $n$ columns). Let $k \leq n$. Write $A$ in the block form $$ A = \begin{pmatrix} A_1 &A_2 \end{pmatrix} $$ where $A_1$ consists of the first $k\times k$ elements and $A_2$ consists of the last $k\times (n-k)$ elements. Now form $n \times n$ matrix $B$ as the following $$ B = \begin{pmatrix} A_1 &A_2 \\ 0 & 1 \end{pmatrix} $$ (here $1$ stands for the identity matrix). This $B$ serves your desired change of variables. Note that if $k<n$ the integral diverges.