Clear[Repeat, vM, vv]; 
    Repeat[vM_, vv_] := 
      Module[{tv, mtv}, tv = vM . vv; 
        mtv = Max[tv];  ((1/mtv)) tv]

mM = {{1, 1}, {1, 2}}

vecs = NestList[Repeat[mM, #] &, {-1, 0}, 4]

Table[
        Show[
Graphics[{
{PointSize[ .02], Blue, Point[{0, 0}]}, 
{Line[{{0, 0}, #}] & /@ 
                  NestList[Repeat[mM, #] &, {-1, 0}, 
                    n]}
}], 
Frame -> True, 
PlotRange -> {{-1.2, 
                1.2}, {-1.2, 
                1.2}}, 
AspectRatio -> 
            Automatic
], {n, 0, 15}];

mM = Table[Random[], {j, 1, 3}, {k, 1, 3}]

Repeat[mM, {1, 1, 1}]

NestList[Repeat[mM, #] &, {1, 1, 1}, 22]

tev = 
        Nest[Repeat[mM, #] &, {1, 1, 1}, 22];
mM . tev/tev

tev = 
        Nest[Repeat[Inverse[mM +  .1 IdentityMatrix[3]], #] &, {1, 1, 1}, 
          132];
mM . tev/tev

tev = 
        Nest[Repeat[Inverse[mM -  .2 IdentityMatrix[3]], #] &, {1, 0, 1}, 
          532];
mM . tev/tev

Eigenvalues[mM]

mM = Table[Random[], {j, 1, 15}, {k, 1, 15}]

tev = 
        Nest[Repeat[mM, #] &, 1  Range[Length[mM]], 
          132];
mM . tev/tev

tev = 
        Nest[Repeat[Inverse[mM - 4 IdentityMatrix[Length[mM]]], #] &, 
          1  Range[Length[mM]], 532];
mM . tev/tev

tev = 
        Nest[Repeat[Inverse[mM - 1 IdentityMatrix[Length[mM]]], #] &, 
          1  Range[Length[mM]], 532];
mM . tev/tev

tev = 
        Nest[Repeat[Inverse[mM -  .4 IdentityMatrix[Length[mM]]], #] &, 
          1  Range[Length[mM]], 532];
mM . tev/tev

tev = 
        Nest[Repeat[Inverse[mM +  .75 IdentityMatrix[Length[mM]]], #] &, 
          1  Range[Length[mM]], 532];
mM . tev/tev

tev = 
        Nest[Repeat[Inverse[mM + 7 IdentityMatrix[Length[mM]]], #] &, 
          1  Range[Length[mM]], 532];
mM . tev/tev

Eigenvalues[mM]