==============================================================

Manipulate[
 m = {{a1, b1}, {a2, b2}}; 
 Show[VectorPlot[m.{x, y}, {x, -10, 10}, {y, -10, 10}, 
   StreamPoints -> {{pt1, pt2, pt3, pt4}}, 
   StreamStyle -> {Red, Thick}, ImageSize -> {460, 310}], 
  Graphics[{Thick, Orange, 
    Map[Line[{-100 #, 100 #}] &, 
     Select[Eigenvectors[
       m], (Im[#[[1]]] == 0 && Im[#[[2]]] == 0) &]]}], 
  PlotLabel -> 
   Row[{Column[{Row[{Column[{Style[
            "\!\(\*OverscriptBox[\(x\), \(.\)]\)", Italic], 
           Style["\!\(\*OverscriptBox[\(y\), \(.\)]\)", Italic]}], 
         Column[{" = ", " = "}], 
         TableForm[m.{Style["x", Italic], Style["y", Italic]}] // 
          N}]}], "  ", 
     Column[{"Eigenvalues:", 
       NumberForm[Chop@N@Eigenvalues[m], {4, 2}]}], "  ", , 
     Column[{"Eigenvectors:", 
       NumberForm[Chop@N@Eigenvectors[m][[1]], {4, 2}], 
       NumberForm[Chop@N@Eigenvectors[m][[2]], {4, 2}] }]}]],
 Style["\!\(\*OverscriptBox[\(\(x\)\(\\\ \)\), \(.\)]\)= \
\!\(\*SubscriptBox[\(a\), \(1\)]\)x + \!\(\*SubscriptBox[\(b\), \
\(1\)]\)y", Bold],
 {{a1, 1, "\!\(\*SubscriptBox[\(a\), \(1\)]\)"}, -10, 10, 
  Appearance -> "Labeled"},
 {{b1, 1, "\!\(\*SubscriptBox[\(b\), \(1\)]\)"}, -10, 10, 
  Appearance -> "Labeled"},
 Style["\!\(\*OverscriptBox[\(y\), \(.\)]\) = \
\!\(\*SubscriptBox[\(a\), \(2\)]\)x + \!\(\*SubscriptBox[\(b\), \
\(2\)]\)y", Bold],
 {{a2, 2, "\!\(\*SubscriptBox[\(a\), \(2\)]\)"}, -10, 10, 
  Appearance -> "Labeled"}, {{b2, -2, 
   "\!\(\*SubscriptBox[\(b\), \(2\)]\)"}, -10, 10, 
  Appearance -> "Labeled"}, {{pt1, {-5.1, 4.8}}, {-10, -10}, {10, 10},
   Locator},
 {{pt2, {4.9, 5.1}}, {-10, -10}, {10, 10}, Locator},
 {{pt3, {-4.9, -5}}, {-10, -10}, {10, 10}, Locator},
 {{pt4, {4.9, -5.2}}, {-10, -10}, {10, 10}, Locator}, 
 TrackedSymbols -> True]