Significance of an eigenvector being equal to a unit vector?

Question

I was reading ahead in my math book when I came across a matrix denoted as A = $\begin{bmatrix} 1 & -1 & 0\\ 2 & -2 & 0\\ 6 & 0 & -2\\ \end{bmatrix}$. I then found the eigensystem for A to be $E_{\lambda=0}=span\begin{bmatrix} 1\\ 1\\ 3\\ \end{bmatrix}, E_{\lambda=-1}=\begin{bmatrix} 1\\ 2\\ 6\\ \end{bmatrix}, E_{\lambda=-2}=\begin{bmatrix} 0\\ 0\\ 1\\ \end{bmatrix}=\hat{k}. $ Is there a significance of $\lambda$ = -2 and its corresponding eigenvector of $\hat{k}$ in relation to A? I ask out of curiosity as I love to try and understand all outcomes in math. Thanks!

Answer 1

Given an eigenvector, any multiple of it is also an eigenvector. You can normalize each eigenvector or not, as you wish. So there is no significance to the magnitude of an eigenvector being $1$.

Answer 2

Yes, if a unit vector is the eigenvector then the corresponding column of the matrix is proportional to that unit vector and vice versa. So if the third column of a matrix is $[0,0,4,0]'$ then the third unit vector is also the eigenvector with eigenvalue equal to 4.