f = {{1, 1/2}, {1/2, 1}}
Eigenvalues[f]
Eigenvectors[f]

<< Graphics`ImplicitPlot`
ImplicitPlot[(x^2) + x*y + (y^2) == 1, {x, -2, 2}, 
      AspectRatio -> Automatic, 
      PlotStyle -> {Thickness[0.02]}]

m = {{0, 1/2}, {1/2, 0}}
Eigenvalues[m]
Eigenvectors[m]

ImplicitPlot[x*y == 1, {x, -3, 3}, 
      AspectRatio -> Automatic, 
      PlotStyle -> {Thickness[0.015]}]