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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.

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