Matrix 1

mA = {{ .8, 0}, {0,  .64}}

MatrixPower[mA, 3] . {3, 3}

ips = {{3, 0}, {3, 3}, {1.5, 3}, {0, 3}, {-1.5, 3}, {-3, 
            3}, {-3, 
            0}, {-3, -3}, {-1.5, -3}, {0, -3}, {1.5, -3}, {3, -3}};

ips = {{3, 3}}

steps = 
        20;



Show[
Graphics[{
{PointSize[
                    0.015], 
                  Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                          k] . #[[1]]], {k, 0, steps}]} & /@ 
              Transpose[{ips, 
                  Range[Length[ips]]}], 
{Thickness[
                    0.002], 
                  Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                          k] . #[[1]], {k, 0, steps}]]} & /@ 
              Transpose[{ips, 
                  Range[Length[
                      ips]]}]
}], 
PlotRange -> {{-3.1, 3.1}, {-3.1, 3.1}}, 
        AspectRatio -> Automatic, 
        Frame -> True
];

Table[
        Show[
Graphics[{
{PointSize[
                      0.015], 
                    Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                            k] . #[[1]]], {k, 0, st}]} & /@ 
                Transpose[{ips, 
                    Range[Length[ips]]}], 
{Thickness[
                      0.002], 
                    Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                            k] . #[[1]], {k, 0, st}]]} & /@ 
                Transpose[{ips, 
                    Range[Length[
                        ips]]}]
}], 
PlotRange -> {{-3.1, 3.1}, {-3.1, 3.1}}, 
          AspectRatio -> Automatic, 
          Frame -> True
], {st, 0, 
          steps}];

Matrix 1a

mA = {{1, 0}, {0,  .8}}

MatrixPower[mA, 3] . {3, 3}

ips = {{3, 0}, {3, 3}, {1.5, 3}, {0, 3}, {-1.5, 3}, {-3, 
            3}, {-3, 
            0}, {-3, -3}, {-1.5, -3}, {0, -3}, {1.5, -3}, {3, -3}};

ips = {{3, 3}}

steps = 
        20;



Show[
Graphics[{
{PointSize[
                    0.015], 
                  Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                          k] . #[[1]]], {k, 0, steps}]} & /@ 
              Transpose[{ips, 
                  Range[Length[ips]]}], 
{Thickness[
                    0.002], 
                  Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                          k] . #[[1]], {k, 0, steps}]]} & /@ 
              Transpose[{ips, 
                  Range[Length[
                      ips]]}]
}], 
PlotRange -> {{-3.1, 3.1}, {-3.1, 3.1}}, 
        AspectRatio -> Automatic, 
        Frame -> True
];

Table[
        Show[
Graphics[{
{PointSize[
                      0.015], 
                    Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                            k] . #[[1]]], {k, 0, st}]} & /@ 
                Transpose[{ips, 
                    Range[Length[ips]]}], 
{Thickness[
                      0.002], 
                    Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                            k] . #[[1]], {k, 0, st}]]} & /@ 
                Transpose[{ips, 
                    Range[Length[
                        ips]]}]
}], 
PlotRange -> {{-3.1, 3.1}, {-3.1, 3.1}}, 
          AspectRatio -> Automatic, 
          Frame -> True
], {st, 0, 
          steps}];

Matrix 1b

Det[{{1, 2}, {3, 4}}]

mA = {{1, 2}, {3, 4}} . {{1, 0}, {0,  .8}} . 
        Inverse[{{1, 2}, {3, 4}}]

MatrixPower[mA, 3] . {3, 3}

ips = {{3, 0}, {3, 3}, {1.5, 3}, {0, 3}, {-1.5, 3}, {-3, 
            3}, {-3, 
            0}, {-3, -3}, {-1.5, -3}, {0, -3}, {1.5, -3}, {3, -3}};

ips = {{3, 3}}

steps = 
        20;



Show[
Graphics[{
{PointSize[
                    0.015], 
                  Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                          k] . #[[1]]], {k, 0, steps}]} & /@ 
              Transpose[{ips, 
                  Range[Length[ips]]}], 
{Thickness[
                    0.002], 
                  Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                          k] . #[[1]], {k, 0, steps}]]} & /@ 
              Transpose[{ips, 
                  Range[Length[
                      ips]]}]
}], 
PlotRange -> {{-3.1, 3.1}, {-3.1, 3.1}}, 
        AspectRatio -> Automatic, 
        Frame -> True
];

Table[
        Show[
Graphics[{
{PointSize[
                      0.015], 
                    Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                            k] . #[[1]]], {k, 0, st}]} & /@ 
                Transpose[{ips, 
                    Range[Length[ips]]}], 
{Thickness[
                      0.002], 
                    Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                            k] . #[[1]], {k, 0, st}]]} & /@ 
                Transpose[{ips, 
                    Range[Length[
                        ips]]}]
}], 
PlotRange -> {{-3.1, 3.1}, {-3.1, 3.1}}, 
          AspectRatio -> Automatic, 
          Frame -> True
], {st, 0, 
          steps}];

Matrix 2

mA = {{1.44, 0}, {0, 1.2}}

MatrixPower[mA, 0] . {3, 3}

steps = 
        20;


ips = {{3, 0}, {3, 3}, {1.5, 3}, {0, 3}, {-1.5, 3}, {-3, 
              3}, {-3, 
              0}, {-3, -3}, {-1.5, -3}, {0, -3}, {1.5, -3}, {3, -3}}/
          10;


Show[
Graphics[{
{PointSize[
                    0.015], 
                  Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                          k] . #[[1]]], {k, 0, steps}]} & /@ 
              Transpose[{ips, 
                  Range[Length[ips]]}], 
{Thickness[
                    0.002], 
                  Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                          k] . #[[1]], {k, 0, steps}]]} & /@ 
              Transpose[{ips, 
                  Range[Length[
                      ips]]}]
}], 
PlotRange -> {{-3.1, 3.1}, {-3.1, 3.1}}, 
        AspectRatio -> Automatic, 
        Frame -> True
];

Table[
        Show[
Graphics[{
{PointSize[
                      0.015], 
                    Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                            k] . #[[1]]], {k, 0, st}]} & /@ 
                Transpose[{ips, 
                    Range[Length[ips]]}], 
{Thickness[
                      0.002], 
                    Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                            k] . #[[1]], {k, 0, st}]]} & /@ 
                Transpose[{ips, 
                    Range[Length[
                        ips]]}]
}], 
PlotRange -> {{-5.1, 5.1}, {-5.1, 5.1}}, 
          AspectRatio -> Automatic, 
          Frame -> True
], {st, 0, 
          steps}];

Matrix 3

mA = {{1.8, 0}, {0,  .64}}

MatrixPower[mA, 0] . {3, 3}

steps = 
        20;


ips = {{ .3, 0}, { .3, 10}, {0.15, 10}, {0, 10}, {- .15, 
            10}, {- .3, 10}, {- .3, 
            0}, {- .3, -10}, {- .15, -10}, {0, -10}, { .15, -10}, { .3, -10}};


Show[
Graphics[{
{PointSize[
                    0.015], 
                  Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                          k] . #[[1]]], {k, 0, steps}]} & /@ 
              Transpose[{ips, 
                  Range[Length[ips]]}], 
{Thickness[
                    0.002], 
                  Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                          k] . #[[1]], {k, 0, steps}]]} & /@ 
              Transpose[{ips, 
                  Range[Length[
                      ips]]}]
}], 
PlotRange -> {{-13.1, 13.1}, {-13.1, 13.1}}, 
        AspectRatio -> Automatic, 
        Frame -> True
];

Table[
        Show[
Graphics[{
{PointSize[
                      0.015], 
                    Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                            k] . #[[1]]], {k, 0, st}]} & /@ 
                Transpose[{ips, 
                    Range[Length[ips]]}], 
{Thickness[
                      0.002], 
                    Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                            k] . #[[1]], {k, 0, st}]]} & /@ 
                Transpose[{ips, 
                    Range[Length[
                        ips]]}]
}], 
PlotRange -> {{-13.1, 13.1}, {-13.1, 13.1}}, 
          AspectRatio -> Automatic, 
          Frame -> True
], {st, 0, 
          steps}];

Matrix 4

mA = {{ .8, 0.5}, {-0.1, 1}}

MatrixPower[mA, 0] . {3, 3}

steps = 
        20;


ips = {{3, 0}, {3, 3}, {1.5, 3}, {0, 3}, {-1.5, 3}, {-3, 
            3}, {-3, 
            0}, {-3, -3}, {-1.5, -3}, {0, -3}, {1.5, -3}, {3, -3}};

    
PR = {{-6.1, 6.1}, {-6.1, 6.1}};

    
Show[
Graphics[{
{PointSize[
                    0.015], 
                  Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                          k] . #[[1]]], {k, 0, steps}]} & /@ 
              Transpose[{ips, 
                  Range[Length[ips]]}], 
{Thickness[
                    0.002], 
                  Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                          k] . #[[1]], {k, 0, steps}]]} & /@ 
              Transpose[{ips, 
                  Range[Length[
                      ips]]}]
}], 
PlotRange -> PR, AspectRatio -> Automatic, 
        Frame -> True
];

Table[
        Show[
Graphics[{
{PointSize[
                      0.015], 
                    Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                            k] . #[[1]]], {k, 0, st}]} & /@ 
                Transpose[{ips, 
                    Range[Length[ips]]}], 
{Thickness[
                      0.002], 
                    Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                            k] . #[[1]], {k, 0, st}]]} & /@ 
                Transpose[{ips, 
                    Range[Length[
                        ips]]}]
}], 
PlotRange -> PR, AspectRatio -> Automatic, 
          Frame -> True
], {st, 0, 
          steps}];

Eigenvalues[mA]

Abs[Eigenvalues[mA]]

Matrix 5 (Example 1)

Eigensystem[{{(1/2), (2/5)}, {- .125, (11/10)}}]

Eigensystem[{{(1/2), (2/5)}, {- p, (11/10)}}]

Example 1

mA = {{(1/2), (2/5)}, {- .104, (11/10)}}

Eigensystem[mA]

MatrixPower[mA, 0] . {3, 3}

steps = 50;
ips = {{3, 0}, {3,  .75}, {3, 1.5}, {3, 2.25}, {3, 3}, {2.25, 
            3}, {1.5, 3}, { .75, 3}, {0, 3}};

    
PR = {{-0.1, 6.1}, {-0.1, 6.1}};

    
Show[
Graphics[{
{PointSize[
                    0.015], 
                  Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                          k] . #[[1]]], {k, 0, steps}]} & /@ 
              Transpose[{ips, 
                  Range[Length[ips]]}], 
{Thickness[
                    0.002], 
                  Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                          k] . #[[1]], {k, 0, steps}]]} & /@ 
              Transpose[{ips, 
                  Range[Length[
                      ips]]}]
}], 
PlotRange -> PR, AspectRatio -> Automatic, 
        Frame -> True
];

Table[
        Show[
Graphics[{
{PointSize[
                      0.015], 
                    Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                            k] . #[[1]]], {k, 0, st}]} & /@ 
                Transpose[{ips, 
                    Range[Length[ips]]}], 
{Thickness[
                      0.002], 
                    Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                            k] . #[[1]], {k, 0, st}]]} & /@ 
                Transpose[{ips, 
                    Range[Length[
                        ips]]}]
}], 
PlotRange -> PR, AspectRatio -> Automatic, 
          Frame -> True
], {st, 0, 
          steps}];

Example 1, different p

mA = {{(1/2), (2/5)}, {p, (11/10)}}

Eigensystem[mA]

mA = {{(1/2), (2/5)}, {-1. , (11/10)}}

Eigensystem[mA]

Abs[Eigensystem[mA][[1, 
        1]]]

steps = 50;
ips = {{3, 0}, {3,  .75}, {3, 1.5}, {3, 2.25}, {3, 3}, {2.25, 
            3}, {1.5, 3}, { .75, 3}, {0, 3}};

    
PR = {{-4.1, 4.1}, {-4.1, 4.1}};

    
Show[
Graphics[{
{PointSize[
                    0.015], 
                  Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                          k] . #[[1]]], {k, 0, steps}]} & /@ 
              Transpose[{ips, 
                  Range[Length[ips]]}], 
{Thickness[
                    0.002], 
                  Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                          k] . #[[1]], {k, 0, steps}]]} & /@ 
              Transpose[{ips, 
                  Range[Length[
                      ips]]}]
}], 
PlotRange -> PR, AspectRatio -> Automatic, 
        Frame -> True
];

Table[
        Show[
Graphics[{
{PointSize[
                      0.015], 
                    Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                            k] . #[[1]]], {k, 0, st}]} & /@ 
                Transpose[{ips, 
                    Range[Length[ips]]}], 
{Thickness[
                      0.002], 
                    Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                            k] . #[[1]], {k, 0, st}]]} & /@ 
                Transpose[{ips, 
                    Range[Length[
                        ips]]}]
}], 
PlotRange -> PR, AspectRatio -> Automatic, 
          Frame -> True
], {st, 0, 
          steps}];

Rotation

mA = 
      0.95 {{Cos[(Pi/3)], -Sin[(Pi/3)]}, {Sin[(Pi/3)], 
            Cos[(Pi/3.) ]}}

Eigensystem[mA]

Abs[Eigensystem[mA][[1, 
        1]]]

steps = 50;
ips = {{3, 0}, {3,  .75}, {3, 1.5}, {3, 2.25}, {3, 3}, {2.25, 
            3}, {1.5, 3}, { .75, 3}, {0, 3}};

    
PR = {{-4.1, 4.1}, {-4.1, 4.1}};

    
Show[
Graphics[{
{PointSize[
                    0.015], 
                  Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                          k] . #[[1]]], {k, 0, steps}]} & /@ 
              Transpose[{ips, 
                  Range[Length[ips]]}], 
{Thickness[
                    0.002], 
                  Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                          k] . #[[1]], {k, 0, steps}]]} & /@ 
              Transpose[{ips, 
                  Range[Length[
                      ips]]}]
}], 
PlotRange -> PR, AspectRatio -> Automatic, 
        Frame -> True
];

Table[
        Show[
Graphics[{
{PointSize[
                      0.015], 
                    Hue[#[[2]]/Length[ips]], Table[Point[MatrixPower[mA, 
                            k] . #[[1]]], {k, 0, st}]} & /@ 
                Transpose[{ips, 
                    Range[Length[ips]]}], 
{Thickness[
                      0.002], 
                    Hue[#[[2]]/Length[ips]], Line[Table[MatrixPower[mA, 
                            k] . #[[1]], {k, 0, st}]]} & /@ 
                Transpose[{ips, 
                    Range[Length[
                        ips]]}]
}], 
PlotRange -> PR, AspectRatio -> Automatic, 
          Frame -> True
], {st, 0, 
          steps}];