When is ${\rm rank}(A+B)={\rm rank}(A)+{\rm rank}(B)$ true?

Question

I am aware of the relation ${\rm rank}(A+B)\leq {\rm rank}(A)+{\rm rank}(B)$, but I am curious of when the equality holds? What are the necessary assumptions for the matrices $A$ and $B$?

Answer 1

Note that the rank is equal to the dimension of the column space (and row space).

$rank(A + B) = \dim col(A + B)$
$\leq \dim (col(A) + col(B))$
$\leq \dim col(A) + \dim col(B)$
$= rank(A) + rank(B)$

Let's think about the span. The column space of (A + B) should contain the column spaces of both A and B. Also, since the rank cannot exceed the amount of rows nor columns, $r \leq m \leq n,$ or $r \leq n \leq m.$ Where m, n represent the rows and columns.

Which means that if $\dim(A) + \dim(B) > n$ or $\dim(A) + \dim(B) > m$, there is certain dependence between the columns of A and B.

Finally
$rank(A + B) = \dim(col(A)) + \dim(col(B)) - \dim(col(A) \cap col(B)) = rank(A) + rank(B)$
If and only if
$\dim(col(A) \cap col(B)) = 0$
Indicating independence.

(i) The column space of (A + B) should contain A and B.
(ii) Vectors in column space of A should be independent from the vectors in column space of B.