Linear algebra



In [1]:

    
import numpy as np



In [2]:

    
np.__version__









    Out[2]:





'1.11.2'

Matrix and vector products

Q1. Predict the results of the following code.



In [61]:

    
x = [1,2]
y = [[4, 1], [2, 2]]
print np.dot(x, y)
print np.dot(y, x)
print np.matmul(x, y)
print np.inner(x, y)
print np.inner(y, x)









    



[8 5]
[6 6]
[8 5]
[6 6]
[6 6]

Q2. Predict the results of the following code.



In [62]:

    
x = [[1, 0], [0, 1]]
y = [[4, 1], [2, 2], [1, 1]]
print np.dot(y, x)
print np.matmul(y, x)









    



[[4 1]
 [2 2]
 [1 1]]
[[4 1]
 [2 2]
 [1 1]]

Q3. Predict the results of the following code.



In [63]:

    
x = np.array([[1, 4], [5, 6]])
y = np.array([[4, 1], [2, 2]])
print np.vdot(x, y)
print np.vdot(y, x)
print np.dot(x.flatten(), y.flatten())
print np.inner(x.flatten(), y.flatten())
print (x*y).sum()

Q4. Predict the results of the following code.



In [65]:

    
x = np.array(['a', 'b'], dtype=object)
y = np.array([1, 2])
print np.inner(x, y)
print np.inner(y, x)
print np.outer(x, y)
print np.outer(y, x)









    



abb
abb
[['a' 'aa']
 ['b' 'bb']]
[['a' 'b']
 ['aa' 'bb']]

Decompositions

Q5. Get the lower-trianglular L in the Cholesky decomposition of x and verify it.



In [97]:

    
x = np.array([[4, 12, -16], [12, 37, -43], [-16, -43, 98]], dtype=np.int32)
L = np.linalg.cholesky(x)
print L
assert np.array_equal(np.dot(L, L.T.conjugate()), x)









    



[[ 2.  0.  0.]
 [ 6.  1.  0.]
 [-8.  5.  3.]]

Q6. Compute the qr factorization of x and verify it.



In [107]:

    
x = np.array([[12, -51, 4], [6, 167, -68], [-4, 24, -41]], dtype=np.float32)
q, r = np.linalg.qr(x)
print "q=\n", q, "\nr=\n", r
assert np.allclose(np.dot(q, r), x)









    



q=
[[-0.85714287  0.39428571  0.33142856]
 [-0.42857143 -0.90285712 -0.03428571]
 [ 0.2857143  -0.17142858  0.94285715]] 
r=
[[ -14.  -21.   14.]
 [   0. -175.   70.]
 [   0.    0.  -35.]]

Q7. Factor x by Singular Value Decomposition and verify it.



In [165]:

    
x = np.array([[1, 0, 0, 0, 2], [0, 0, 3, 0, 0], [0, 0, 0, 0, 0], [0, 2, 0, 0, 0]], dtype=np.float32)
U, s, V = np.linalg.svd(x, full_matrices=False)
print "U=\n", U, "\ns=\n", s, "\nV=\n", v
assert np.allclose(np.dot(U, np.dot(np.diag(s), V)), x)









    



U=
[[ 0.  1.  0.  0.]
 [ 1.  0.  0.  0.]
 [ 0.  0.  0. -1.]
 [ 0.  0.  1.  0.]] 
s=
[ 3.          2.23606801  2.          0.        ] 
V=
[[ 1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0.  1.]]

Matrix eigenvalues

Q8. Compute the eigenvalues and right eigenvectors of x. (Name them eigenvals and eigenvecs, respectively)



In [68]:

    
x = np.diag((1, 2, 3))
eigenvals = np.linalg.eig(x)[0]
eigenvals_ = np.linalg.eigvals(x)
assert np.array_equal(eigenvals, eigenvals_)
print "eigenvalues are\n", eigenvals
eigenvecs = np.linalg.eig(x)[1]
print "eigenvectors are\n", eigenvecs









    



eigenvalues are
[ 1.  2.  3.]
eigenvectors are
[[ 1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0.  1.]]

Q9. Predict the results of the following code.



In [69]:

    
print np.array_equal(np.dot(x, eigenvecs), eigenvals * eigenvecs)









    



True

Norms and other numbers

Q10. Calculate the Frobenius norm and the condition number of x.



In [12]:

    
x = np.arange(1, 10).reshape((3, 3))
print np.linalg.norm(x, 'fro')
print np.linalg.cond(x, 'fro')









    



16.8819430161
4.56177073661e+17

Q11. Calculate the determinant of x.



In [22]:

    
x = np.arange(1, 5).reshape((2, 2))
out1 = np.linalg.det(x)
out2 = x[0, 0] * x[1, 1] - x[0, 1] * x[1, 0]
assert np.allclose(out1, out2)
print out1

Q12. Calculate the rank of x.



In [35]:

    
x = np.eye(4)
out1 = np.linalg.matrix_rank(x)
out2 = np.linalg.svd(x)[1].size
assert out1 == out2
print out1

Q13. Compute the sign and natural logarithm of the determinant of x.



In [49]:

    
x = np.arange(1, 5).reshape((2, 2))
sign, logdet = np.linalg.slogdet(x)
det = np.linalg.det(x)
assert sign == np.sign(det)
assert logdet == np.log(np.abs(det))
print sign, logdet









    



-1.0 0.69314718056

Q14. Return the sum along the diagonal of x.



In [57]:

    
x = np.eye(4)
out1 = np.trace(x)
out2 = x.diagonal().sum()
assert out1 == out2
print out1

4.0

Solving equations and inverting matrices

Q15. Compute the inverse of x.



In [60]:

    
x = np.array([[1., 2.], [3., 4.]])
out1 = np.linalg.inv(x)
assert np.allclose(np.dot(x, out1), np.eye(2))
print out1









    



[[-2.   1. ]
 [ 1.5 -0.5]]



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