Question

Let u and v be two non-zero N-dimensional complex column vectors. Show that the rank of the N by N matrix uv^(conjugate transpose) is one.

Answer 1

Question 23818 Let $u$ and $v$ be two non-zero $N$-dimensional complex column vectors. Show that the rank of the $N \times N$ matrix $u\overline{v}'$ is one.

Solution. We know the following general inequality $\mathrm{rank}(u\overline{v}') \leq \min\{\mathrm{rank}(u), \mathrm{rank}(\overline{v}')\} = 1$, since $u$ and $\overline{v}'$ are non-zero. Next $\mathrm{rank}(u\overline{v}') \geq 1$, since $u$ and $\overline{v}'$ are non-zero, consequently

$$\mathrm{rank}(u\overline{v}') = 1$$