The first secret is that eigenvectors are not really the point; the fundamental concept is eigenspaces, and the point of eigenvectors is to furnish bases for eigenspaces. Eigenspaces are unique, but you have to make choices to pick out a basis of eigenvectors.
An eigenspace for a linear operator T:V→V is a subspace W⊆V which is T-invariant (meaning that TW⊆W) and on which T acts as multiplication by some scalar λ (the “eigenvalue”). You would love to have such a thing if it existed, because multiplication by a scalar is super simple to understand (for example, it’s super easy to compute powers of T acting on W since you just take powers of λ). But there’s no reason a priori to expect that such things exist.
In other words, most of the time you can split V up into a bunch of directions such that T acts by scaling each of those directions (this intuition is complicated by the fact that we’re working over a field like C but as a first pass this isn’t bad). I want to underline that this is a miracle; there was no reason a priori that linear operators on finite-dimensional vector spaces should behave this simply, and in the infinite-dimensional case things are much more complicated.
I don’t know if that addressed any of your confusions; as I said above, more specific questions are better.
The first secret is that eigenvectors are not really the point; the fundamental concept is eigenspaces, and the point of eigenvectors is to furnish bases for eigenspaces. Eigenspaces are unique, but you have to make choices to pick out a basis of eigenvectors.
An eigenspace for a linear operator T:V→V is a subspace W⊆V which is T-invariant (meaning that TW⊆W) and on which T acts as multiplication by some scalar λ (the “eigenvalue”). You would love to have such a thing if it existed, because multiplication by a scalar is super simple to understand (for example, it’s super easy to compute powers of T acting on W since you just take powers of λ). But there’s no reason a priori to expect that such things exist.
The miracle is that if V is finite-dimensional and we’re working over an algebraically closed field such as C, then T always has a nontrivial eigenspace, and in fact most of the time V is the direct sum of the eigenspaces of T. (In general V is the direct sum of the generalized eigenspaces of T; this is a slightly weaker version of the existence of Jordan normal form.)
In other words, most of the time you can split V up into a bunch of directions such that T acts by scaling each of those directions (this intuition is complicated by the fact that we’re working over a field like C but as a first pass this isn’t bad). I want to underline that this is a miracle; there was no reason a priori that linear operators on finite-dimensional vector spaces should behave this simply, and in the infinite-dimensional case things are much more complicated.
I don’t know if that addressed any of your confusions; as I said above, more specific questions are better.