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.

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