I'm working with a kind of "truncating" matrix function $\tau_r:M_{n\times n}\to M_{n\times r}$, where $r\leq n$, defined by $\tau_r(A)=B$, where $b_{ij}=a_{ij}$ for $j\leq r$.
Is this a well known function in matrix theory, or is there a similar one I could use, without having to define my own function $\tau$ ? Or maybe there exists a widely used notation instead ?
I haven't encountered such truncating function, but you're basically getting a submatrix of a matrix. So let me say the following.
In Horn & Johnson's Matrix Analysis, they introduced the notation: Given $A\in M_{n}$, if $\alpha,\beta\subseteq\{1,\ldots,n\}$, then $A[\alpha, \beta]$ denotes the submatrix of entries that lie in the rows of $A$ indexed by $\alpha$ and the columns indexed by $\beta$.
You might want to write your matrix as $A[\{1,\ldots, n\}, \{1,\ldots, r\}]$. It's pretty tedious to write this though. It doesn't hurt to introduce some notation like $A(n,r)$ or $A_r$ if you have to use it repeatedly.