# 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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