On sum of matrices

Question

Suppose we have a matrix $M\in\Bbb Q^{n\times n}$ with no $0$ elements and we write as sum of two matrices $M_1$ and $M_2$ on following constraint.

$M_{1,ij}=M_{ij}$ and $M_{2,ij}=0$ or $M_{1,ij}=0$ and $M_{2,ij}=M_{ij}$ or $M_{1,ij}=M_{2,ij}=\frac{M_{ij}}2$.

If $M$ is of rank $1$ under what conditions on the split can we expect ranks of $M_1$ and $M_2$ to be bound by $O(\log\|M\|_2)$?

Answer 1

Take $M$ with all $M_{ij} = 1$, $M_1 = I$, $M_2 = M-I$. Since the eigenvalues of $M$ are $0$ and $n$, $M_1$ and $M_2$ both have rank $n$ if $n > 1$.

In the other direction, if $M_1 = M_2 = M/2$, $\text{rank}(M_1) = \text{rank}(M_2) = \text{rank}(M) = 1$.