0.7 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.15 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\}.

  1. You can prove this for an (n\times n) matrix A by considering the set \{|v\rangle,A|v\rangle,A^2|v\rangle,\ldots,A^n|v\rangle\} of vectors in \mathbb{C}^n. Since this has n+1 elements, it must be linearly dependent, and so (after some lengthy algebra) we can construct an eigenpair.↩︎