Eigenvectors and matrices

Question

If A is a matrix of rank 1, show that any nonzero vector in the image of A is an eigenvector of A.

I feel like this is easy but I am stuck. I am stuck on figuring out the image of A.

Answer 1

Note that the image of $A$ is one dimensional (since $A$ has rank $1$). So, there is some vector $u$ such that $Av$ is always a (scalar) multiple of $u$ for any vector $v$.

Now, why is $u$ an eigenvector?

Answer 2

Since $dim(Im)=1$, $Im=<v>$, where $Im$ is the image of $A$ and $v$ is a nonzero vector.

Take $u \in Im$, $u=\alpha v$ with $\alpha \neq 0$. We know that $Au \in Im$, so $Au=\beta v$.

Therefore, $Au=\beta\alpha ^{-1}u$