Why is a matrix multiplied by an eigenvector not parallel to that eigenvector?

Question

If $\lambda$ is a non-zero eigenvalue with a corresponding eigenvector $v$, then $A v$ is parallel to $v$.

This statement is false. Why is that? Would it be parallel to $\lambda v$?

Answer 1

It's hard to know what to make of your question as you didn't specify what matrix $\vec{v}$ is an eigenvector of. If $A$ were the matrix corresponding to that eigenvector, then $A\vec{v}=\lambda\vec{v}$, by definition, meaning the statement would be true. Since you're sure the statement is false, then we can assume that $A$ is not the matrix corresponding to that eigenvector. In that case, of course, $A\vec{v}$ need not equal $\lambda \vec{v}$. If you're confused about what it means to be parallel or what an Eigenvector is, I recommend 3b1b's video on the topic.