Singular value decomposition | Wikipedia audio article
Channel: wikipedia tts
Date: 06/05/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Singular_value_decomposition


00:04:31 1 Statement of the theorem
00:06:23 2 Intuitive interpretations
00:06:35 2.1 Rotation, scaling
00:07:38 2.2 Singular values as semiaxis of an ellipse or ellipsoid
00:08:07 2.3 The columns of iU/i and iV/i are orthonormal bases
00:08:15 2.4 Geometric meaning
00:08:55 3 Example
00:09:54 4 SVD and spectral decomposition
00:11:32 4.1 Singular values, singular vectors, and their relation to the SVD
00:14:50 4.2 Relation to eigenvalue decomposition
00:16:10 5 Applications of the SVD
00:34:50 5.1 Pseudoinverse
00:35:03 5.2 Solving homogeneous linear equations
00:40:34 5.3 Total least squares minimization
00:43:53 5.4 Range, null space and rank
00:44:38 5.5 Low-rank matrix approximation
00:45:13 5.6 Separable models
00:46:08 5.7 Nearest orthogonal matrix
00:46:20 5.8 The Kabsch algorithm
00:47:35 5.9 Signal processing
00:47:52 5.10 Other examples
00:49:20 6 Existence proofs
00:49:43 6.1 Based on the spectral theorem
00:49:55 6.2 Based on variational characterization
00:51:24 7 Calculating the SVD
00:53:50 7.1 Numerical approach
00:57:53 7.2 Analytic result of 2 × 2 SVD
00:58:33 8 Reduced SVDs
01:00:25 8.1 Thin SVD
01:00:30 8.2 Compact SVD
01:00:40 8.3 Truncated SVD
01:01:09 9 Norms
01:01:19 9.1 Ky Fan norms
01:01:37 9.2 Hilbert–Schmidt norm
01:06:21 10 Variations and generalizations
01:09:03 10.1 Mode-ik/i representation
01:10:11 10.2 Tensor SVD
01:35:27 10.3 Scale-invariant SVD
01:41:34 10.4 HOSVD of functions – numerical reconstruction – TP model transformation
01:42:55 10.5 Bounded operators on Hilbert spaces
01:43:07 10.6 Singular values and compact operators
01:49:12 11 History
01:56:50 12 See also
01:57:31 13 Notes
01:58:52 14 References
02:00:08 15 External links
02:01:57 Norms
02:02:06 Ky Fan norms
02:04:03 Tr[(M* M)½] (the eigenvalues of M* M are the squares of the singular values).
02:04:16 Hilbert–Schmidt norm
02:07:09 Variations and generalizations
02:07:21 Mode-k representation
02:08:58 Tensor SVD
02:10:11 Scale-invariant SVD
02:11:39 HOSVD of functions – numerical reconstruction – TP model transformation
02:12:25 Bounded operators on Hilbert spaces
02:15:31 Singular values and compact operators
02:19:16 History
02:22:36 See also



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SUMMARY
=======
In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix.  It is the generalization of the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any 
  
    
      
        m
        ×
        n
      
    
    {\displaystyle m\times n}
   matrix via an extension of the polar decomposition.  It has many useful applications in signal processing and statistics.
Formally, the singular-value decomposition of an 
  
    
      
        m
        ×
        n
      
    
    {\displaystyle m\times n}
   real or complex matrix 
  
    
      
        
          M
        
      
    
    {\displaystyle \mathbf {M} }
   is a factorization of the form 
  
    
      
        
          U
          Σ
          
            V
            
              ∗
            
          
        
      
    
    {\displaystyle \mathbf {U\Sigma V^{*}} }
  , where 
  
    
      
        
          U
        
      
    
    {\displaystyle \mathbf {U} }
   is an 
  
    
      
        m
        ×
        m
      
    
    {\displaystyle m\times m}
   real or complex unitary matrix, 
  
    
      
        
          Σ
        
      
    
    {\displaystyle \mathbf {\Sigma } }
   is an 
  
    
      
        m
        ×
        n
      
    
    {\displaystyle m\times n}
   rectangular diagonal matrix with non-negative real numbers on the diagonal, and 
  
    
      
        
          V
        
      
 ...