Eigenvalues and Eigenvectors
Introduction
Ax is a linear transformation: A maps the input vector x to the output vector b.
For some linear transformations (i.e. some, but not all, matrices A), there are certain input vectors x where the output vector b is also just a linear transformation of the input vector x. In other words, A mapped x to a scaled version of x. That scaling factor is notated λ.
For example, rotation around the y axis in 3 dimensions by θ degrees is calculated with:
┌ ┐ │ cos(θ) 0 sin(θ)│ │ 0 1 0│ │ -sin(θ) 0 cos(θ)| └ ┘
The y axis, and more importantly any vector that only moves on the y axis, will not be transformed by this A. (And because the linear transformation involves no scaling, λ = 1.)
The certain input vectors are the eigenvectors of A. The scaling factors are the eigenvalues of A.
Definition
Eigenvalues and eigenvectors are the pairs of λ and x that satisfy Ax = λx and |A - λI| = 0.
Unless A is defective, there should be n unique pairs of eigenvectors and eigenvalues. If there is a repeated eigenvalue, there may not be n independent eigenvectors.
Properties
Adding nI to A does not change its eigenvectors and adds n to the eigenvalues.
The sum of the eigenvalues is the trace (sum of diagonal). The product of the eigenvalues is the determinant.