Eigenvalue | Wikipedia audio article
Channel: wikipedia tts
Date: 20/03/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors


00:01:56 1 Overview
00:06:33 2 History
00:09:22 3 Eigenvalues and eigenvectors of matrices
00:13:57 3.1 Eigenvalues and the characteristic polynomial
00:15:44 3.2 Algebraic multiplicity
00:16:13 3.3 Eigenspaces, geometric multiplicity, and the eigenbasis for matrices
00:16:24 3.4 Additional properties of eigenvalues
00:17:57 3.5 Left and right eigenvectors
00:20:27 3.6 Diagonalization and the eigendecomposition
00:20:44 3.7 Variational characterization
00:22:27 3.8 Matrix examples
00:30:16 3.8.1 Two-dimensional matrix example
00:33:43 3.8.2 Three-dimensional matrix example
00:33:45 3.8.3 Three-dimensional matrix example with complex eigenvalues
00:35:50 3.8.4 Diagonal matrix example
00:39:32 3.8.5 Triangular matrix example
00:40:25 3.8.6 Matrix with repeated eigenvalues example
00:40:34 4 Eigenvalues and eigenfunctions of differential operators
00:42:12 4.1 Derivative operator example
00:42:37 5 General definition
00:43:08 5.1 Eigenspaces, geometric multiplicity, and the eigenbasis
00:43:45 5.2 Zero vector as an eigenvector
00:44:18 5.3 Spectral theory
00:44:48 5.4 Associative algebras and representation theory
00:45:00 6 Dynamic equations
00:48:00 7 Calculation
00:53:42 7.1 Classical method
00:54:41 7.1.1 Eigenvalues
00:56:10 7.1.2 Eigenvectors
00:57:20 7.2 Simple iterative methods
00:58:46 7.3 Modern methods
01:00:03 8 Applications
01:00:56 8.1 Eigenvalues of geometric transformations
01:02:03 8.2 Schrödinger equation
01:03:33 8.3 Molecular orbitals
01:04:33 8.4 Geology and glaciology
01:07:30 8.5 Principal component analysis
01:08:36 8.6 Vibration analysis
01:09:30 8.7 Eigenfaces
01:10:10 8.8 Tensor of moment of inertia
01:15:06 8.9 Stress tensor
01:15:27 8.10 Graphs
01:15:50 8.11 Basic reproduction number
01:18:13 9 See also
01:21:17 10 Footnotes
01:23:13 11 Notes
01:24:07 12 References
01:24:17 13 External links
01:25:45 13.1 Theory
01:29:15 13.2 Demonstration applets
01:30:18 Geology and glaciology
01:33:49 Principal component analysis
01:35:48 Vibration analysis
01:39:32 Eigenfaces
01:40:51 Tensor of moment of inertia
01:41:17 Stress tensor
01:41:46 Graphs
01:44:51 Basic reproduction number
01:46:15 See also
01:46:48 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 ...