Say that we have n complex-valued vectors ${\mathbf{z}}_{i}$ and we want to evaluate:

${\Vert \sum _{i=1}^{n}{\mathbf{z}}_{i}\Vert}_{2}^{2}$

Now, I know that for real vectors ${\mathbf{x}}_{i}$ it holds:

${\Vert \sum _{i=1}^{n}{\mathbf{x}}_{i}\Vert}_{2}^{2}=\sum _{i=1}^{n}{\Vert {\mathbf{x}}_{i}\Vert}_{2}^{2}+\sum _{i\ne j}{\mathbf{x}}_{i}\cdot {\mathbf{x}}_{j}$

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?

${\Vert \sum _{i=1}^{n}{\mathbf{z}}_{i}\Vert}_{2}^{2}$

Now, I know that for real vectors ${\mathbf{x}}_{i}$ it holds:

${\Vert \sum _{i=1}^{n}{\mathbf{x}}_{i}\Vert}_{2}^{2}=\sum _{i=1}^{n}{\Vert {\mathbf{x}}_{i}\Vert}_{2}^{2}+\sum _{i\ne j}{\mathbf{x}}_{i}\cdot {\mathbf{x}}_{j}$

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?