Eigenvalues and eigenvectors
Given an operator A, an eigenvector is a non-zero vector |v\rangle such that
A|v\rangle = \lambda|v\rangle
for some \lambda\in\mathbb{C} (which is called the corresponding eigenvalue).
We call the pair (\lambda,|v\rangle) an eigenpair, and we call the set of eigenvalues the spectrum of A, denoted by \sigma(A).
It is a surprising (but incredibly useful) fact that every operator has at least one eigenpair.
Geometrically, an eigenvector of an operator A is a vector upon which A simply acts by “stretching”.
Rewriting the defining property of an eigenpair (\lambda,|v\rangle), we see that
(A-\lambda\mathbf{1})|v\rangle = 0
which tells us that the operator A-\lambda\mathbf{1} has a non-zero kernel, and is thus non-invertible.
This gives a useful characterisation of the spectrum in terms of a determinant:
\sigma(A) = \{\lambda\in\mathbb{C} \mid \det(A-\lambda\mathbf{1})=0\}.