Decomposition of the One-Norm

Question

Why can I write:

$||\beta||_1 = z^{T}\beta$

(where $||\cdot||_1$ is the one-norm)

where $z^{T}$ is either $sgn(b_{j})$ if $b_{j} \ne 0$ or $[-1,1]$ if $b_{j} = 0$

Answer 1

$||\beta||_1 =\sum_{i\in I} |b_i|$ and $z^{T}\beta=\sum_{i\in I} z_ib_i$ so you want $|b_j|=z_jb_j$ for each $j\in I$.

Now if $b_j=0$, then $|b_j|=0=z_jb_j$, so $z_j$ is a free choice.

If $b_j\neq 0$, then $z_j=\frac{|b_j|}{b_j}=\text{sgn}(b_j)$.