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== Introduction ==

'''''A'''x'' 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'''''.

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'''Eigenvalues''' and '''eigenvectors''' are paired. For an [[LinearAlgebra/MatrixInversion|invertible]] ''n'' x ''n'' matrix, there should be ''n'' pairs. They satisfy the conditions '''''A'''x = λx'' and ''|'''A''' - λ'''I'''| = 0''. '''Eigenvalues''' and '''eigenvectors''' are the pairs of ''λ'' and ''x'' that satisfy '''''A'''x = λx'' and ''|'''A''' - λ'''I'''| = 0''.
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If there is a repeated eigenvalue, there may not be ''n'' independent eigenvectors. 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.

Eigenvalues and Eigenvectors

Contents

  1. Eigenvalues and Eigenvectors
    1. Introduction
    2. Definition
      1. Properties

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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LinearAlgebra/EigenvaluesAndEigenvectors (last edited 2024-02-06 05:04:28 by DominicRicottone)