P=rand(4)

P =

    0.8147    0.6324    0.9575    0.9572
    0.9058    0.0975    0.9649    0.4854
    0.1270    0.2785    0.1576    0.8003
    0.9134    0.5469    0.9706    0.1419

sum(P,2)

ans =

    3.3618
    2.4536
    1.3634
    2.5727

s=sum(P,2)

s =

    3.3618
    2.4536
    1.3634
    2.5727

Q=diag(1./s)*P

Q =

    0.2424    0.1881    0.2848    0.2847
    0.3692    0.0398    0.3933    0.1978
    0.0931    0.2043    0.1156    0.5870
    0.3550    0.2126    0.3773    0.0551

sum(Q,2)

ans =

    1.0000
    1.0000
    1.0000
    1.0000

sum(Q)

ans =

    1.0597    0.6447    1.1709    1.1247

sum(Q,2)

ans =

    1.0000
    1.0000
    1.0000
    1.0000

Q=diag(s)\P

Q =

    0.2424    0.1881    0.2848    0.2847
    0.3692    0.0398    0.3933    0.1978
    0.0931    0.2043    0.1156    0.5870
    0.3550    0.2126    0.3773    0.0551

P=randstoc(4)

P =

    0.1749    0.2719    0.2814    0.2718
    0.4870    0.0190    0.4030    0.0910
    0.2563    0.2748    0.2405    0.2285
    0.4140    0.4030    0.1692    0.0137

[V,D]=eig(P')

V =

  Columns 1 through 2

   0.6117             0.6170          
   0.4614             0.0539          
   0.5467            -0.7778          
   0.3378             0.1069          

  Columns 3 through 4

   0.6102             0.6102          
  -0.1227 + 0.4750i  -0.1227 - 0.4750i
   0.0200 - 0.3277i   0.0200 + 0.3277i
  -0.5074 - 0.1473i  -0.5074 + 0.1473i


D =

  Columns 1 through 2

   1.0000                  0          
        0            -0.0340          
        0                  0          
        0                  0          

  Columns 3 through 4

        0                  0          
        0                  0          
  -0.2590 + 0.1415i        0          
        0            -0.2590 - 0.1415i

v=V(:,1)'

v =

    0.6117    0.4614    0.5467    0.3378

v=v/sum(v)

v =

    0.3125    0.2357    0.2793    0.1726

sum(v)

ans =

    1.0000

v*P

ans =

    0.3125    0.2357    0.2793    0.1726

v

v =

    0.3125    0.2357    0.2793    0.1726

P*[1;1;1;1]

ans =

    1.0000
    1.0000
    1.0000
    1.0000

D

D =

  Columns 1 through 2

   1.0000                  0          
        0            -0.0340          
        0                  0          
        0                  0          

  Columns 3 through 4

        0                  0          
        0                  0          
  -0.2590 + 0.1415i        0          
        0            -0.2590 - 0.1415i

P

P =

    0.1749    0.2719    0.2814    0.2718
    0.4870    0.0190    0.4030    0.0910
    0.2563    0.2748    0.2405    0.2285
    0.4140    0.4030    0.1692    0.0137

w=invariantPotenze(P,1e-6,100)

w =

    0.3125    0.2357    0.2793    0.1726

v

v =

    0.3125    0.2357    0.2793    0.1726

A=P'-I
??? Undefined function or variable 'I'.

A=P'-eye(4)

A =

   -0.8251    0.4870    0.2563    0.4140
    0.2719   -0.9810    0.2748    0.4030
    0.2814    0.4030   -0.7595    0.1692
    0.2718    0.0910    0.2285   -0.9863

b=zeros(4,1)

b =

     0
     0
     0
     0

A\b
Warning: Matrix is close to singular or badly
scaled.
         Results may be inaccurate. RCOND =
         2.913631e-17.

ans =

     0
     0
     0
     0

x=[A;ones(1,4)]\[b;1]

x =

    0.3125
    0.2357
    0.2793
    0.1726

x'

ans =

    0.3125    0.2357    0.2793    0.1726

v

v =

    0.3125    0.2357    0.2793    0.1726

w

w =

    0.3125    0.2357    0.2793    0.1726

v=[P
v=([P'-eye(size(P));ones(1,length(P))]\[zeros(length(P),1);1])'

v =

    0.3125    0.2357    0.2793    0.1726

diary off