quit
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%  Finding a stable distribution
%%  Problem 30, p. 412
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A = [.69 .16 .2;.12 .74 .14;.19 .1 .66]

A =

    0.6900    0.1600    0.2000
    0.1200    0.7400    0.1400
    0.1900    0.1000    0.6600

B = A-eye(3)

B =

   -0.3100    0.1600    0.2000
    0.1200   -0.2600    0.1400
    0.1900    0.1000   -0.3400

C = [1 1 1 1;B(1,:) 0;B(2,:) 0; B(3,:) 0]

C =

    1.0000    1.0000    1.0000    1.0000
   -0.3100    0.1600    0.2000         0
    0.1200   -0.2600    0.1400         0
    0.1900    0.1000   -0.3400         0

rref(C)

ans =

    1.0000         0         0    0.3661
         0    1.0000         0    0.3317
         0         0    1.0000    0.3022
         0         0         0         0

%%  Therefore the stable distribution is approximately
%%  x = .3611
%%  y = .3317
%%  z = .3022
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%  We could also have ignored the last row of C 
%%  because we know that Gauss-Jordan will yield a zero row.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

C1 = C(1:3,:)

C1 =

    1.0000    1.0000    1.0000    1.0000
   -0.3100    0.1600    0.2000         0
    0.1200   -0.2600    0.1400         0

%%  Note that the MATLAB command says to take rows 1 to 3, and all 
%%  columns of C

rref(C1)

ans =

    1.0000         0         0    0.3661
         0    1.0000         0    0.3317
         0         0    1.0000    0.3022

%%  This yields the same stable distribution.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%  Finally we could find the stable distribution
%%  by taking higher and higher powers of the transition
%%  matrix A
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A

A =

    0.6900    0.1600    0.2000
    0.1200    0.7400    0.1400
    0.1900    0.1000    0.6600

A^5

ans =

    0.3908    0.3433    0.3614
    0.2997    0.3879    0.3087
    0.3095    0.2688    0.3299

A^10

ans =

    0.3674    0.3645    0.3664
    0.3289    0.3364    0.3299
    0.3036    0.2992    0.3037

A^15

ans =

    0.3662    0.3660    0.3662
    0.3315    0.3321    0.3315
    0.3023    0.3019    0.3023

A^20

ans =

    0.3661    0.3661    0.3661
    0.3317    0.3317    0.3317
    0.3022    0.3021    0.3022

A^25

ans =

    0.3661    0.3661    0.3661
    0.3317    0.3317    0.3317
    0.3022    0.3022    0.3022

%%  For any higher power the columns will not change
%%  up to 4 decimal places

A^100

ans =

    0.3661    0.3661    0.3661
    0.3317    0.3317    0.3317
    0.3022    0.3022    0.3022

%%  Each column is the same as the stable distribution found
%%  earlier.

quit