Linear Algebra - rank question

Question

consider two vectors $v,w$ is exist $R$. Let $A = vv^T+ww^T$. What is the maximum possible rank of $A$?

I know that the rank of $v$ and rank of $v^T$ is same. And also, rank of $w$ and $w^T$ is same.

But is it possible that i can add the rank? For example, $v$ has $n$ ranks, and $w$ has $m$ ranks, $r(A) = n + m$ ranks? I cannot solve the above question.

Answer 1

One way to approach this question is to note that $$ vv^T + ww^T = \pmatrix{v & w} \pmatrix{v&w}^T. $$ Alternatively, it is generally true that $\operatorname{rank}(A + B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$.

Answer 2

The operation $\mathbf{v}\mathbf{v}^T$ is an outer product. It is known that the outer product matrix has rank 1, since the columns are all proportional to the first column, therefore they are linearly dependent. As the same goes for $\mathbf{w}\mathbf{w}^T$, the maximum rank of the sum of two outer products is 2 (and possibly less).