If two matrices with pure nonzero complex entries
in $M_n( \Bbb{C}) $
are of the same rank, then can we say that both matrices are similar?
Edit:
Where the term 'pure nonzero complex entries' indicates that $\forall \ a_{i,j}$ entry in $A \in M_n( \Bbb{C} ), a_{i,j} \ $ has nonzero imaginary part. Where $i$ runs from $1,2 \cdots ,n$ , and so is $j$
Best Answer
No. Take, for instance,$$A=\begin{bmatrix}2i&i\\-2i&-i\end{bmatrix}\text{ and }B=\begin{bmatrix}i&i\\-i&-i\end{bmatrix}.$$Both of them have rank $1$. But they are not similar. For instance, $A^2\neq0$, but $B^2=0$.