Rank of real/imaginary part of a complex matrix of rank 1

Question

$\newcommand{\rank}{\operatorname{rank}}$

Given a complex matrix $M \in \mathbb{C}^{p \times q}$ of rank $1$.

It appears that $\rank \Re ( M ) \leq 2$ and $\rank \Im ( M ) \leq 2$. If this is true, how to actually prove that?

Answer 1

Hint : Prove that $\overline{M}$ also has rank $1$, and then use the fact that $\Re ( M )=\frac{M+\overline{M}}{2}$ and $\Im( M )=\frac{M-\overline{M}}{2i}$.

Answer 2

Let $A:= \Re ( M )$ and $B:= \Im ( M )$, hence $M=A+iB$. We define$\newcommand{\rank}{\operatorname{rank}}$

$\overline{M}:= A-iB$. Then $\rank(\overline{M})=1$. Thus

$\rank(2A)= \rank(M+\overline{M}) \le 1+1=2$. Hence $\rank(A) \le 2$.

Similar: $\rank(B) \le 2$