I have this system $\nabla b(z_1,\ldots,z_m)=\psi^\prime(\theta)\nabla\theta(z_1,\ldots,z_m)$. Where we have
$\theta(z_1,\ldots,z_m)=\sum_i^m z_i$ and $b(z_1,\ldots,z_m)$=$\sum_i^m z_i^2$
What is the expression of $\psi^\prime(\theta)$ in terms of $z$.
Solution: $\|(2z_1,\ldots,2z_m)^t\|=|\psi^\prime(\theta)|\|(1,\ldots,1)^t\|$
Implies
$|\psi^\prime(\theta)|=\frac{4\sqrt{\sum z_i^2}}{\sqrt{m}}$
What is the sign of $\psi'(\theta)$?
Can I say this
$\text{sign}(\psi'(\theta))=\text{sign}(2z_i)$, for all $i$. That will mean the sign of all $z_i$s are the same. But this is not true.