Eigenvalues[{{1,1},{0,-1}}]

Eigenvectors[{{1,1},{0,-1}}]

p7=VectorPlot[{x+y,-y},{x,-6,6},{y,-6,6},AxesLabel->{"(Subscript[x, 1])","(Subscript[x, 2])"},Axes->True]

dy7[x0_,y0_]:=NDSolve[{x'[t]==x[t]+y[t],y'[t]==- y[t],x[0]==x0,y[0]==y0},{x[t],y[t]},{t,-5,10}];

ratk[1]=dy7[1,1];ratk[2]=dy7[-1,1];ratk[3]=dy7[1,2];ratk[4]=dy7[-1,2];ratk[5]=dy7[1,3];ratk[6]=dy7[-1,3];ratk[7]=dy7[1,1/2];
ratk[8]=dy7[1,-1];ratk[9]=dy7[-1,-1];ratk[10]=dy7[1,-2];ratk[11]=dy7[-1,-2];ratk[12]=dy7[1,-3];ratk[13]=dy7[-1,-3];ratk[14]=dy7[1,-1/2];

p7b=ParametricPlot[Evaluate[Table[{x[t],y[t]}/.ratk[i],{i,14}]],{t,-2,10}, PlotRange->{{-6,6},{-6,6}}, PlotPoints->100,AxesLabel->{"(Subscript[x, 1])","(Subscript[x, 2])"},PlotStyle->Thickness[0.005]]

Show[{p7,p7b,Graphics[{PointSize[.02],Point[{1,1}]}],Graphics[{PointSize[.02],Point[{-1,2}]}]}, PlotRange->{{-6,6},{-6,6}}]