I'v read this: Do non-square matrices have eigenvalues?
Where @Any explains that "If $A$ is non-square, then $A:\mathbb{R}^m\rightarrow \mathbb{R}^n$, where $m\neq n$. Hence $Av=\lambda v$ makes no sense, since $Av\notin\mathbb{R}^m$".
I understand that a non-square matrix can't satisfy the mathematical definition for eigenvectors/eigenvalues.
However, I always thought the motivation behind eigenvectors was to find the vectors that keep pointing in the same direction after a linear transformation, regardless of scaling.
In that sense, the vectors $<4, 5, 0>$ and $<4, 5>$ seem to follow that idea, despite the difference in dimensionality. After all, they point in exactly the same direction.
I know that a lot of mathematical concepts are defined out of convenience, as tools. I'm assuming Eigenvectors were defined because there is a lot of value in being able to identify vectors that keep pointing in the same direction after a transformation.
Does that mean there is no value in identifying the vectors that remain pointing in the same direction if there is a change of dimensionality?
If there is no such value, why not?
And If there is value, what is that value and what tools have been developed to label and identify such vectors.
