Using vector operations to construct vector $c$ such that $c_i = ||a_i + b_i||_2$ without using the diagonal operator

Question

Given two vectors $a=[a_1,\cdots,a_N]$ and $b=[b_1,\cdots,b_N]$

I would like to construct another vector $c=[c_1,\cdots,c_N]$ via vector operation of $a$ and $b$ where $c_i = ||a_i + b_i||_2$.

Is it possible to do so?

Answer 1

Your vector $c$ does not only depend on the vectors $a$ and $b$, but also on the choosen orthonormal basis $(e_i)_{1\leq i\leq n}$. In addition one has $c_i\geq0$ in any case. The only thing that comes to mind is writing $$c=\sum_{i=1}^n \sqrt{(a\cdot e_i)^2+(b\cdot e_i)^2}\>e_i\ .$$