Let $A\in\mathbb{R}^{m\times m} $ and $B\in\mathbb{R}^{m\times m}$. It is known that $rank(A+B)\leq rank(A)+rank(B)$. My question is when does equality holds for this inequality? What is the condition for the case of more than 2 matrices?
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There are necessary conditions. For example, see this question. In general, the matrix sum is not well-behaved with respect to rank. Consider $0=A+(-A)$. So there is no nice necessary and sufficient condition. – Dietrich Burde Apr 05 '17 at 09:15
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In that case, "if the equality holds, then ... " I am looking for possibly not an answer but an insight since as you mentioned matrix sum is not well-behaved wrt rank which results with the equality not a result of the equality. – DnzSnl Apr 05 '17 at 09:30
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The insight is indeed to consider the column spaces, as in the above MSE question. – Dietrich Burde Apr 05 '17 at 09:32