A more general soft constraint is the Tikhonov regularization constraint $ \mathbf{w}^\text{T}\Gamma^\text{T}\Gamma\mathbf{w} \leq C $ which can capture relationships among the $w_i$ (the matrix $\Gamma$ is the Tikhonov regularizer).
(a) What should $\Gamma$ be to obtain a constraint of the form $\sum_{q=0}^Q w_q^2 \leq C$?
I think this is just the identity matrix since $\sum_{q=0}^Q w_q^2 = \mathbf{w}^\text{T}\mathbf{w}$
(b) What should $\Gamma$ be to obtain a constraint of the form $\left(\sum_{q=0}^Q w_q\right)^2 \leq C$?
To me, this is saying $\mathbf{ww} \leq C$. How is it possible to get $\mathbf{ww} = \mathbf{w}^\text{T}\mathbf{w}$ just by multiplying by some $\Gamma^\text{T}\Gamma$? Where am I going wrong here?