Conceptual question about equivalence of eigenvectors Suppose for a matrix the eigenvalue is 1 and the eigenvector is (2,-3). Then does that mean (-4,6) and (4,-6) are equivalent eigenvectors as the ratios are the same? 

Let $\vec{x} = [2, -3]^T$ be an eigenvector of the matrix $A$ with eigenvalue $\lambda$. Then
$A \vec{x} = \lambda \vec{x}$.
The vector you are suggesting to look at is $\vec{y} = [4, -6]^T = 2 \vec{x}$. So note that
$A \vec{y} = 2A \vec{x} = 2 \lambda \vec{x} = \lambda \vec{y}$,
so $\vec{y}$ is also an eigenvector corresponding to the same eigenvalue.
In general, any constant multiple would be, and also if $\vec{x},\vec{y}$ are eigenvectors corresponding to the same eigenvalue, so would be $a \vec{x} + b\vec{y}$ for any $a,b \in \mathbb{R}$.
