Introduction to Computer Science An by ert554898

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									An Example for Illustrating
    Maximization of
   Rayleigh’s Quotient



       Shyh-Kang Jeng
Department of Electrical Engineering/
Graduate Institute of Communication/
Graduate Institute of Networking and
             Multimedia
                                        1
       Rayleigh’s Quotient

         x' Bx
f ( x) 
          x' x
x  x1 x2 '
    1        
     2      0
                             1        1
B  a        , a  b, 1  2 , 2  2
    0       1             b        a
    
            2
            b 
                                          2
     Rayleigh’s Quotient in
      Polar Coordinates

         2
        x1     2
              x2    b 2 cos 2   a 2 sin 2 
                  2
                                                 
x' Bx  2  2                                  
                               2 2              
       a   b                  a b               
x' x  x1  x2   2
        2    2

x' Bx b cos   a sin 
          2    2       2   2
            2 2
 x' x       a b

                                                     3
Equation of the Ellipse in Polar
         Coordinates
   2        2
  x1       x2
   2
           2
                1
  a        b
                                          1
   2 
         b 2 cos 2   a 2 sin 2    
                                    
                    2 2              
                   a b               
  x' Bx      1
          2       *2
   x' x     
                                               4
  Inversion with Respect to
       the Unit Circle

 *  1   x2
                       


            *
                  x1




                              5
   Ellipse and Its Inversion with
    Respect to the Unit Circle

                   90        2                                     90        2
             120                 60                          120                 60
                          1.5                                             1.5

       150               1             30              150               1             30

                         0.5                                             0.5


 180                                         0   180                                         0




       210                             330             210                             330



             240                 300                         240                 300
                   270                                             270




a = 2, b = 0.5
                                                                                                 6
Maximum and Minimum of
  Rayleigh’s Quotient


  a  2, b  0.5
        x' Bx          1
  max          1     2
                            4
  x  0 x' x           b
        x' Bx           1
  min          2          0.25
  x  0 x' x           a2


                                     7

								
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