# example of turning vector field
var('x','y')

# matrix eigenvalue/eigenvector pairs are -1, [1,0] and -2, [1,-1/2]
M = matrix([[-1,2],[0,-2]])
system = M*vector([x,y])
M.eigenvalues()
M.eigenvectors_right()

# plot vector field and eigenvectors ("eigendirections")
p = plot_vector_field(M*vector([x,y]),(x,-1.6,1.6),(y,-1.6,1.6),aspect_ratio=1)
p+= plot(-1/2*x,(x,-1.6,1.6),color="red")
p+= plot(0*x,(x,-1.6,1.6),color="red")

# show trajectories from a bunch of initial conditions
t = srange(0,50,.1)
for ic in zip(srange(-1,1,.2),srange(-1,1,.2)):
    sol = desolve_odeint(M*vector([x,y]),dvars=[x,y],ics=ic,times=t)
    p += list_plot(sol,plotjoined=True)
show(p)