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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 can we expect ranks of $M_1$ and $M_2$ to be bound by $O(1)$ or $O(\log\|M\|_2)$?

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 $O(\log\|M\|_2)$?

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 $1$ then can we say anything about worst case and best case ranks of $M_1$ and $M_2$ and conditions on which both happen?

Under what conditions can we expect ranks to be bound by $O(1)$ or $O(\log\|M\|_2)$?

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 can we expect ranks of $M_1$ and $M_2$ to be bound by $O(1)$ or $O(\log\|M\|_2)$?

Suppose we have a matrix $M\in\Bbb Q^{n\times n}$ and we write as sum of two matrices $M_1$ and $M_2$ on following constraints.

1. If $M_{ij}=0$ then either $M_{1,ij}=M_{2,ij}=0$ or $M_{1,ij}=-M_{2,ij}=4^{k_{ij}}\neq1$ where $k_{ij}\in\Bbb Z$.

2. If $M_{ij}\not=0$ then we have $M_{1,ij}=M_{ij}$ 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 $1$ then can we say anything about worst case and best case ranks of $M_1$ and $M_2$ and conditions on which both happen?

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 $1$ then can we say anything about worst case and best case ranks of $M_1$ and $M_2$ and conditions on which both happen?

Under what conditions can we expect ranks to be bound by $O(1)$ or $O(\log\|M\|_2)$?