-->// Already diagonal but with a negative value
 
-->a=[4 0; 0 -9]
 a  =
 
    4.    0.  
    0.  - 9.  
 
-->[u s v]=svd(a)
 v  =
 
    0.    1.  
  - 1.    0.  
 s  =
 
    9.    0.  
    0.    4.  
 u  =
 
    0.    1.  
    1.    0.  
 
-->u*s*v
 ans  =
 
  - 4.    0.  
    0.    9.  
 
-->u*s*v'
 ans  =
 
    4.    0.  
    0.  - 9.  
 
-->// complex values
 
-->a=[%i 0; 0 -%i]
 a  =
 
    i      0    
    0    - i    
 
-->[u s v]=svd(a)
 v  =
 
  - 1.    0   
    0   - 1.  
 s  =
 
    1.    0.  
    0.    1.  
 u  =
 
  - i      0    
    0      i    
 
-->u*s*v'
 ans  =
 
    i      0    
    0    - i    
 
-->// non-symmetric
 
-->p=[1 1; 0 1]
 p  =
 
    1.    1.  
    0.    1.  
 
-->a=p*[4 0 ; 0 9 ] *p^(-1)
 a  =
 
    4.    5.  
    0.    9.  
 
-->[u s v]=svd(a)
 v  =
 
    0.2075915  - 0.9782156  
    0.9782156    0.2075915  
 s  =
 
    10.499728    0.         
    0.           3.4286602  
 u  =
 
    0.5449135  - 0.8384922  
    0.8384922    0.5449135  
 
-->// note lack of relation between eigenvalues and singular values
 
-->u*s*v'
 ans  =
 
    4.           5.  
  - 1.355E-17    9.  
 
-->// svd works with non-square matrices
 
-->a=[1 0; 1 1 ; 0 1]
 a  =
 
    1.    0.  
    1.    1.  
    0.    1.  
 
-->[u s v]=svd(a)
 v  =
 
  - 0.7071068    0.7071068  
  - 0.7071068  - 0.7071068  
 s  =
 
    1.7320508    0.  
    0.           1.  
    0.           0.  
 u  =
 
  - 0.4082483    0.7071068    0.5773503  
  - 0.8164966  - 1.160E-17  - 0.5773503  
  - 0.4082483  - 0.7071068    0.5773503  
 
-->u*s*v'
 ans  =
 
    1.           1.014E-17  
    1.           1.         
    1.672E-16    1.         
 
-->[p, d] = spec (a'*a)
 d  =
 
    1.    0.  
    0.    3.  
 p  =
 
  - 0.7071068    0.7071068  
    0.7071068    0.7071068  
 
-->diary(0)