>> A = [2, 1, 0, 0; 1, 2, 1, 0; 0, 1, 2, 1; 0, 0, 1, 2]
A =

   2   1   0   0
   1   2   1   0
   0   1   2   1
   0   0   1   2


Zjistim uhel, o ktery je potreba zrotovat 1. sloupec, aby byl snulovan
subdiagonalni prvek.

>> alpha = atan(A(2,1)/A(1,1))
alpha =  1.1071


Vytvorim transformaci rotace 

>> c = cos(alpha)
c =  0.44721

>> s = sin(alpha)
s =  0.89443

>> ROT1 = [c,s,0,0;-s,c,0,0;0,0,1,0;0,0,0,1]
ROT1 =

   0.89443   0.44721   0.00000   0.00000
  -0.44721   0.89443   0.00000   0.00000
   0.00000   0.00000   1.00000   0.00000
   0.00000   0.00000   0.00000   1.00000


>> A1 = ROT1 * A
A1 =

   2.23607   1.78885   0.44721   0.00000
   0.00000   1.34164   0.89443   0.00000
   0.00000   1.00000   2.00000   1.00000
   0.00000   0.00000   1.00000   2.00000

Pokracujeme dalsi sloupcem.


>> alpha = atan(A1(3,2)/A1(2,2))
alpha =  0.64052

>> c = cos(alpha)
c =  0.80178

>> s = sin(alpha)
s =  0.59761

>> ROT2 = [1, 0,0,0; 0, c,s,0;0,-s,c,0;0,0,0,1]
ROT2 =

   1.00000   0.00000   0.00000   0.00000
   0.00000   0.80178   0.59761   0.00000
   0.00000  -0.59761   0.80178   0.00000
   0.00000   0.00000   0.00000   1.00000


>> A2 = ROT2 * A1
A2 =

   2.23607   1.78885   0.44721   0.00000
   0.00000   1.67332   1.91237   0.59761
   0.00000  -0.00000   1.06904   0.80178
   0.00000   0.00000   1.00000   2.00000


pokracuji 3 sloupcem:

>> alpha = atan(A2(4,3)/A2(3,3))
alpha =  0.75204
>> s = sin(alpha)
s =  0.68313
>> c = cos(alpha)
c =  0.73030
>> ROT3 = [1, 0,0,0; 0, 1,0,0;0,0,c,s;0,0,-s,c]
ROT3 =

   1.00000   0.00000   0.00000   0.00000
   0.00000   1.00000   0.00000   0.00000
   0.00000   0.00000   0.73030   0.68313
   0.00000   0.00000  -0.68313   0.73030

>> R = ROT3 * A2
R =

   2.23607   1.78885   0.44721   0.00000
   0.00000   1.67332   1.91237   0.59761
   0.00000  -0.00000   1.46385   1.95180
   0.00000   0.00000   0.00000   0.91287

>> Q = (ROT3 * ROT2 * ROT1)'
Q =

   0.89443  -0.35857   0.19518  -0.18257
   0.44721   0.71714  -0.39036   0.36515
   0.00000   0.59761   0.58554  -0.54772
   0.00000   0.00000   0.68313   0.73030


>> Q * R
ans =

   2.00000   1.00000   0.00000  -0.00000
   1.00000   2.00000   1.00000   0.00000
   0.00000   1.00000   2.00000   1.00000
   0.00000   0.00000   1.00000   2.00000


Krok QR-algoritmu bez Wilkinsonova posunu

>>  Q' * A * Q
ans =

   2.80000   0.74833  -0.00000   0.00000
   0.74833   2.34286   0.87482   0.00000
   0.00000   0.87482   2.19048   0.62361
   0.00000   0.00000   0.62361   0.66667