MatrixForm[{20,30,8}]

{20,30,8}//MatrixForm

v={20,30,8}

v

0.5 v//MatrixForm

m={{0.1,0.2,0.3},{0.5,0.7,0.3},{0.1,0.3,0.8}}

MatrixForm[m]

MatrixForm[v]

m.v//MatrixForm

20 0.1+30 0.2+8 0.3

IdentityMatrix[3]//MatrixForm

IdentityMatrix[3].v

m={{0,1,3,2},{0.5,0,0,0},{0,0.8,0,0},{0,0,0.5,0}};

MatrixForm[m]

v={20,30,10,8};

m.v//MatrixForm

m.m.v//MatrixForm

This is written as m.m.v=(m^2).v
To do this in Mathematica, use MatrixPower[]

MatrixPower[m,2]//MatrixForm

MatrixPower[m,2].v//MatrixForm

MatrixPower[m,20].v//MatrixForm

{v,MatrixPower[m,1].v,MatrixPower[m,2].v}

Table[MatrixPower[m,i].v,{i,0,2}]

Transpose[Table[MatrixPower[m,i].v,{i,0,2}]]

ListLinePlot[Transpose[Table[MatrixPower[m,i].v,{i,0,9}]],PlotRange->All]

ListLinePlot[Log[Transpose[Table[MatrixPower[m,i].v,{i,0,11}]]],PlotRange->All]

Join[{Total[a]},a]

a=Transpose[Table[MatrixPower[m,i].v,{i,0,11}]];
b=Join[{Total[a]},a];
p1=ListLinePlot[Log[b],PlotRange->All,PlotStyle->Prepend[ColorData[97,"ColorList"],Black]]

Eigenvalues[m]

Re[Eigenvalues[m]][[1]]

f[x_]:=1.29177 x

Total[v]

p2=ListPlot[Log[NestList[f,68,11]]]

Show[p1,p2]

MatrixPower[m,200].v/Total[MatrixPower[m,200].v]

c=Eigenvectors[m][[1]]

Total[c]

c/Total[c]//Re