Rank of the difference of matrices Let $A$ and $B$ be to $n \times n$ matrices. My question is:
Is $\operatorname{rank}(A-B) \geq \operatorname{rank}(A) - \operatorname{rank}(B)$ true in general? Or maybe under certain assumptions?

Set $X=A-B$ and $Y=B$. You are asking whether $\operatorname{rank}(X) + \operatorname{rank}(Y) \ge \operatorname{rank}(X+Y)$. This is true in general. Let $W=\operatorname{Im}(X)\cap\operatorname{Im}(Y)$. Let $U$ be a complementary subspace of $W$ in $\operatorname{Im}(X)$ and $V$ be a complementary subspace of $W$ in $\operatorname{Im}(Y)$. Then we have $\operatorname{Im}(X)=U+W$ and $\operatorname{Im}(Y)=V+W$ by definition and also $\operatorname{Im}(X+Y)\subseteq U+V+W$. Therefore
$$\operatorname{rank}(X) + \operatorname{rank}(Y) = \dim U + \dim V + 2\dim W\ge \dim U+\dim V+\dim W \ge\operatorname{rank}(X+Y).$$
