Here is problem.

Let's say we have two $m \times n $ matrices $A$ and $B$.

I want to find the minimum $m+n$ such that

there exists $v_1,\cdots,v_m \in \mathbb{R}^m$ and $u_1,\cdots,u_n \in \mathbb{R}^n$ and $a_1,\cdots,a_m \in \mathbb{R}^n$ and $a_1',\cdots,a_m' \in \mathbb{R}^n$ and $b_1,\cdots,b_n \in \mathbb{R}^m$ and $b_1',\cdots,b_n' \in \mathbb{R}^m$ satisfying

$A+ \sum_i v_ia_i^T + \sum_j b_iu_i^T = 0$

and

$B+ \sum_i v_ia_i'^T + \sum_j b_i'u_i^T= 0$.

The answer has to greater or equal to $max_{\lambda_1,\lambda_2} rank(\lambda_1 A + \lambda_2 B)$. Is this bound tight?