If $xy^T$ is symmetric then $x$ and $y$ are linearly dependent.

Question

I am working on matrices over $\mathbb Z_n$. And I need to show that $xy^T$ is symmetric if and only if $x$ and $y$ are linearly dependent. The other direction can be easily verified. However, I find it difficult to prove that if $xy^T$ is symmetric then $x$ and $y$ are linearly dependent.

Answer 1

If $xy^T$ is symmetric we have that $xy^T = yx^T$. Now apply both sides to an arbitary vector $v$ such that $x^Tv \neq 0$:

$$xy^Tv = yx^Tv \to \lambda_1 x = \lambda_2 y$$

Hence the vectors are linearly dependent.