# coding: utf-8

# # Rank-1 Approximation

# In[3]:

import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt


# In[67]:

np.random.seed(17)
n = 10
X = np.random.randn(2, n)

X[1] = 0.7*X[0] + 0.2 * X[1]

# uncomment this for a different data set
#X[0] = 0.1 * X[0]


# In[47]:

pt.figure(figsize=(6,6))
pt.gca().set_aspect("equal")
pt.xlim([-2, 2])
pt.ylim([-2, 2])
pt.grid()
pt.plot(X[0], X[1], "o")


# Now compute the SVD. Use `numpy.linalg.svd(..., full_matrices=False)`.

# In[52]:

U, sigma, VT = la.svd(X, full_matrices=False)


# Now find the vectors `u` and `v`:

# In[53]:

u = U[:, 0]
v = VT[0]


# Now find `X1`:

# In[61]:

X1 = np.outer(u, v) * sigma[0]


# In[66]:


pt.figure(figsize=(6,6))

pt.arrow(0, 0, u[0], u[1], lw=3)

pt.gca().set_aspect("equal")
pt.xlim([-2, 2])
pt.ylim([-2, 2])
pt.grid()

pt.plot(X[0], X[1], "ob", label="X")
pt.plot(X1[0], X1[1], "og", label="X1")
pt.legend(loc="best")


# * Is this the same as least-squares data fitting?
# * Would least-squares data fitting deal with the second data set above?