Eigenvalue, eigenvector and eigenspace | Wikipedia audio article
Channel: wikipedia tts
Date: 29/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors


00:02:18 1 Overview
00:07:44 2 History
00:11:01 3 Eigenvalues and eigenvectors of matrices
00:16:28 3.1 Eigenvalues and the characteristic polynomial
00:18:35 3.2 Algebraic multiplicity
00:19:07 3.3 Eigenspaces, geometric multiplicity, and the eigenbasis for matrices
00:19:21 3.4 Additional properties of eigenvalues
00:21:12 3.5 Left and right eigenvectors
00:24:08 3.6 Diagonalization and the eigendecomposition
00:24:29 3.7 Variational characterization
00:26:31 3.8 Matrix examples
00:35:50 3.8.1 Two-dimensional matrix example
00:39:55 3.8.2 Three-dimensional matrix example
00:39:58 3.8.3 Three-dimensional matrix example with complex eigenvalues
00:42:23 3.8.4 Diagonal matrix example
00:46:45 3.8.5 Triangular matrix example
00:47:45 3.8.6 Matrix with repeated eigenvalues example
00:47:55 4 Eigenvalues and eigenfunctions of differential operators
00:49:53 4.1 Derivative operator example
00:50:21 5 General definition
00:50:59 5.1 Eigenspaces, geometric multiplicity, and the eigenbasis
00:51:43 5.2 Zero vector as an eigenvector
00:52:23 5.3 Spectral theory
00:52:59 5.4 Associative algebras and representation theory
00:53:13 6 Dynamic equations
00:56:47 7 Calculation
01:03:40 7.1 Classical method
01:04:50 7.1.1 Eigenvalues
01:06:36 7.1.2 Eigenvectors
01:07:58 7.2 Simple iterative methods
01:09:40 7.3 Modern methods
01:11:14 8 Applications
01:12:17 8.1 Eigenvalues of geometric transformations
01:13:34 8.2 Schrödinger equation
01:15:20 8.3 Molecular orbitals
01:16:29 8.4 Geology and glaciology
01:20:05 8.5 Principal component analysis
01:21:21 8.6 Vibration analysis
01:22:23 8.7 Eigenfaces
01:23:08 8.8 Tensor of moment of inertia
01:29:04 8.9 Stress tensor
01:29:27 8.10 Graphs
01:29:52 8.11 Basic reproduction number
01:32:36 9 See also
01:36:14 10 Footnotes
01:38:31 11 Notes
01:39:32 12 References
01:39:42 13 External links
01:41:24 13.1 Theory
01:45:33 13.2 Demonstration applets
01:46:46 Geology and glaciology
01:50:54 Principal component analysis
01:53:11 Vibration analysis
01:57:33 Eigenfaces
01:59:02 Tensor of moment of inertia
01:59:31 Stress tensor
02:00:05 Graphs
02:03:42 Basic reproduction number
02:05:18 See also
02:05:55 Footnotes



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SUMMARY
=======
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and v is a vector in V that is not the zero vector, then v is an eigenvector of T if T(v) is a scalar multiple of v.  This condition can be written as the equation

  
    
      
        T
        (
        
          v
        
        )
        =
        λ
        
          v
        
        ,
      
    
    {\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} ,}
  where λ is a scalar in the field F, known as the eigenvalue,  characteristic value, or characteristic root associated with the eigenvector v.
If the vector space V is finite-dimensional, then the linear transformation T can be represented as a square matrix A, and the vector v by a column vector, rendering the above mapping as a matrix multiplication on the left-hand side and a scaling of the column vector on the right-hand side in the equation

  
    
      
        A
        
          v
        
        =
        λ
        
          v
        
        .
      
    
    {\displaystyle A\mathbf {v} =\lambda \mathbf {v} .}
  There is a direct correspondence between n-by-n square matrices and linear transformations from an n-dimensional vector space to itself, given any basis of the vector space. For this reason, it is equivalent to define eigenvalues and eigenvectors using e ...