Either works when trying to find the eigenvalues, but only the former works when trying to find corresponding eigenvectors. I can understand how it makes a difference, but what I don't understand how one is supposed to "know" the former is the "correct" form, since it starts from here:
$AX = λX$
And from there, you can end up with either:
$AX - λX = 0$ or $λX - AX = 0$
And finally:
$(A-λI)X = 0$ or $(λI-A)X = 0$
And so should they not both be correct? Furthermore, once it's in that form, can you not multiply both sides by $-1$ to flip them?
I guess I'm overlooking some sort of rule of algebra when dealing with matrices. Let me know. Thanks.
EDIT
Here is an example:
$\pmatrix{5&3\\6&2}$ has an eigenvalue $-4$
If we do $(A-λI)X = 0$: $\pmatrix{9&3\\6&2}*X = 0$
This solves to $X = \pmatrix{-1\\3}$ (and multiples of it)
If we do $(λI-A)X = 0$: $\pmatrix{-9&3\\6&-2}*X = 0$
This solves to $X = \pmatrix{1\\3}$ (and multiples of it)