Calculate 1-norm of a vector using another matrix or vector

Question

Let's say I have a vector a. I would like to construct a matrix or vector b such that if I multiply a and b, I get the 1-norm of a.

In other words I would like to calculate the 1-norm of a vector by multiplying it with a matrix or a vector. a*b=|a|

Is such a matrix/vector possible to construct, assuming that I do not know anything about a?

Answer 1

No. A matrix specifies a linear operator, and you can clearly see that norms are not linear - for example, the norm of $a+(-a)=0$ is zero, but $||a||+||-a|| \neq 0$ unless $a$ is zero.

However, if $a$ has non-negative components, note that $1^t a = ||a||$.