import scipy.linalg as lin
var('t')
def rotell(sig,umat,t,offset=0):
temp = matrix(umat)*matrix(2,1,[sig[0]*cos(t),sig[1]*sin(t)])
return [offset+temp[0][0],temp[1][0]]
@interact
def svd_vis(a11=slider(-1,1,.05,1),a12=slider(-1,1,.05,1),a21=slider(-1,1,.05,0),a22=slider(-1,1,.05,1),ofs= ('offset image from domain',False)):
rf_low = RealField(12)
my_mat = matrix(rf_low,2,2,[a11,a12,a21,a22])
u,s,vh = lin.svd(my_mat.numpy())
if ofs:
offset = 3
fsize = 6
colors = [(1,0,0),(0,0,1),(1,0,0),(0,0,1)]
else:
offset = 0
fsize = 5
colors = [(1,0,0),(0,0,1),(.7,.2,0),(0,.3,.7)]
vvects = sum([arrow([0,0],matrix(vh).row(i),rgbcolor = colors[i]) for i in (0,1)])
uvects = Graphics()
for i in (0,1):
if s[i] != 0: uvects += arrow([offset,0],vector([offset,0])+matrix(s*u).column(i),rgbcolor = colors[i+2])
html('<h3>Singular value decomposition: image of the unit circle and the singular vectors</h3>')
html("$A = %s = %s %s %s$"%(latex(my_mat), latex(matrix(rf_low,u.tolist())), latex(matrix(rf_low,2,2,[s[0],0,0,s[1]])), latex(matrix(rf_low,vh.tolist()))))
image_ell = parametric_plot(rotell(s,u,t, offset),(0,2*pi))
graph_stuff=circle((0,0),1)+image_ell+vvects+uvects
graph_stuff.set_aspect_ratio(1)
show(graph_stuff,frame = False,axes=False,figsize=[fsize,fsize])
