Solving equation with outer product of sum of vectors

Question

Consider two finite M-dimensional complex vectors $\bf{u},\bf{v}\in \mathbb {C}^M$, and $\mathbf{A}=(\bf{u}+\bf{v})(\bf{u}+\bf{v})^\dagger\in\mathbb{C}^{M\times M}$.

Given $\bf{u}\in \mathbb {C}^M$ and $\mathbf{A}\in\mathbb{C}^{M\times M}$, how do I extract $\bf{v}\in \mathbb {C}^M$ ?

Answer 1

Let $\mathbf w:=\mathbf u+\mathbf v$. Observe, that every column of $A$ is a multiple of $\mathbf w$ and $\operatorname{trace}(\mathbf A)=\|\mathbf w\|^2$. Hence, we get $$\mathbf w = \pm\frac{\sqrt{\operatorname{trace}(\mathbf A)}}{\|\mathbf A\mathbf e_1\|}\cdot \mathbf A\mathbf e_1. $$ Now, you can get $\mathbf v=\mathbf w-\mathbf u$.