How do you know which eigenvectors are which column? Does it matter? So in my example if I found the eigenvectors how would I know it wouldn't be

This matrix isn't diagonal so it cannot be in our case , and if the first eigenvalue appears in D then the first column of the invertible matrix P will be an eigenvector corresponding to this eigenvalue...

Tonio
Is it because we start with the smallest values of

and work our way up and the eigenvectors found get put into the matrix D left from right? If this is true, then what happens if 2 eigenvectors are associated with one eigenvalue (is this even possible)?