najdu vlastni hodnoty leve dolni 2x2 submatice

>> A([3,4],[3,4])
ans =

   2   1
   1   2


>> eig(A([3,4],[3,4]))
ans =

   1
   3

Vezmu tu hodnotu ktera je (v absolutni hodnote) blize levemu dolnimu rohu
(ted je to nastejno, vezmu 1)

>> Ashifted = A- 1* eye(4)
Ashifted =

   1   1   0   0
   1   1   1   0
   0   1   1   1
   0   0   1   1

Shiftnutou matici

>> [Q,R]=qr(Ashifted)
Q =

  -0.70711   0.00000  -0.40825   0.57735
  -0.70711   0.00000   0.40825  -0.57735
  -0.00000  -1.00000   0.00000   0.00000
  -0.00000  -0.00000   0.81650   0.57735

R =

  -1.41421  -1.41421  -0.70711   0.00000
   0.00000  -1.00000  -1.00000  -1.00000
   0.00000   0.00000   1.22474   0.81650
   0.00000   0.00000   0.00000   0.57735

vynasobím v opačném pořadí a znovu 

>> A1 = R * Q + 1*eye(4)
A1 =

   3.00000   0.70711   0.00000  -0.00000
   0.70711   2.00000  -1.22474   0.00000
   0.00000  -1.22474   1.66667   0.47140
   0.00000   0.00000   0.47140   1.33333

>> eig(A1([3,4],[3,4]))
ans =

   1
   2

>> Ashifted = A1 - 2* eye(4)
Ashifted =

   1.00000   0.70711   0.00000  -0.00000
   0.70711   0.00000  -1.22474   0.00000
   0.00000  -1.22474  -0.33333   0.47140
   0.00000   0.00000   0.47140  -0.66667

>> [Q,R]=qr(Ashifted)
Q =

  -0.81650   0.18257  -0.47809  -0.26726
  -0.57735  -0.25820   0.67612   0.37796
  -0.00000  -0.94868  -0.27603  -0.15430
  -0.00000   0.00000  -0.48795   0.87287

R =

  -1.22474  -0.57735   0.70711   0.00000
   0.00000   1.29099   0.63246  -0.44721
   0.00000   0.00000  -0.96609   0.19518
   0.00000   0.00000   0.00000  -0.65465

>> A2 = R * Q + 2*eye(4)
A2 =

   3.33333  -0.74536  -0.00000  -0.00000
  -0.74536   1.06667   0.91652  -0.00000
   0.00000   0.91652   2.17143   0.31944
   0.00000   0.00000   0.31944   1.42857


... jeste jednu iteraci

>> eig(A2([3,4],[3,4]))
ans =

   2.2899
   1.3101

>> Ashifted = A2 - 1.3101 * eye(4)
Ashifted =

   2.02323  -0.74536  -0.00000  -0.00000
  -0.74536  -0.24343   0.91652  -0.00000
   0.00000   0.91652   0.86133   0.31944
   0.00000   0.00000   0.31944   0.11847

>> [Q,R]=qr(Ashifted)
Q =

  -0.93835  -0.16197  -0.29449  -0.08086
   0.34569  -0.43966  -0.79939  -0.21950
  -0.00000   0.88344  -0.45182  -0.12406
  -0.00000   0.00000  -0.26479   0.96431

R =

  -2.15616   0.61525   0.31683  -0.00000
   0.00000   1.03744   0.35798   0.28220
   0.00000   0.00000  -1.20640  -0.17570
   0.00000   0.00000   0.00000   0.07461

>> A2 = R * Q + 1.3101 *eye(4)
A2 =

   3.54602   0.35863  -0.00000   0.00000
   0.35863   1.17024  -1.06578   0.00000
   0.00000  -1.06578   1.90170  -0.01976
   0.00000   0.00000  -0.01976   1.38205

Vidime ze mimodiagonalni prvky v poslednim radku (a poslednim sloupci)
se postupne zmensuji... az budou dostatecne male, vezmeme prvek z leveho 
horniho rohu, prohlasime ho za vlastni hodnotu. 
Algoritmus pokracuje pro matici s odstranenym poslednim radkem a poslenim sloupcem.