Say that we have $n$ complex-valued vectors $\mathbf{z}_{i}$ and we want to evaluate:
$$\left\|\sum_{i=1}^{n} \mathbf{z}_{i}\right\|_2^{2}$$
Now, I know that for real vectors $\mathbf{x}_{i}$ it holds:
$$\left\|\sum_{i=1}^{n} \mathbf{x}_{i}\right\|_2^{2}=\sum_{i=1}^{n}\left\|\mathbf{x}_{i}\right\|_2^{2}+\sum_{i \neq j} \mathbf{x}_{i} \cdot \mathbf{x}_{j}$$
But when dealing with complex vectors, the last term with the inner product will be a complex number in general so I am not sure that this formula generalizes for this case. What am I missing?
Best Answer
Generalizing @enzotib's answer to $n$ terms, since $\Vert z\Vert_2^2=z^\ast\cdot z$ we have$$\left\Vert\sum_iz_i\right\Vert_2^2=\sum_iz_i^\ast\cdot\sum_jz_j=\sum_{ij}z_i^\ast\cdot z_j=\sum_iz_i^\ast\cdot z_i+\sum_{i\ne j}z_i^\ast\cdot z_j.$$