Linear Algebra – Solutions¶
These solutions demonstrate NumPy’s linear algebra capabilities. Each solution shows the primary function and, where applicable, alternative approaches. The assert statements verify correctness by checking mathematical identities (e.g., that QR decomposition reconstructs the original matrix).
import numpy as np
np.__version__
Matrix and Vector Products¶
Solutions demonstrating np.dot(), np.matmul(), np.inner(), np.outer(), and np.vdot() with various input shapes.
Q1. Predict the results of the following code.
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)
Q2. Predict the results of the following code.
x = [[1, 0], [0, 1]]
y = [[4, 1], [2, 2], [1, 1]]
print np.dot(y, x)
print np.matmul(y, x)
Q3. Predict the results of the following code.
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.
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)
Decompositions¶
Solutions for Cholesky (np.linalg.cholesky), QR (np.linalg.qr), and SVD (np.linalg.svd) decompositions, each verified by reconstructing the original matrix.
Q5. Get the lower-trianglular L in the Cholesky decomposition of x and verify it.
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)
Q6. Compute the qr factorization of x and verify it.
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)
Q7. Factor x by Singular Value Decomposition and verify it.
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)
Matrix Eigenvalues¶
Solutions using np.linalg.eig() and np.linalg.eigvals() to compute eigenvalues and eigenvectors.
Q8. Compute the eigenvalues and right eigenvectors of x. (Name them eigenvals and eigenvecs, respectively)
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
Q9. Predict the results of the following code.
print np.array_equal(np.dot(x, eigenvecs), eigenvals * eigenvecs)
Norms and Other Numbers¶
Solutions using np.linalg.norm(), np.linalg.cond(), np.linalg.det(), np.linalg.matrix_rank(), np.linalg.slogdet(), and np.trace().
Q10. Calculate the Frobenius norm and the condition number of x.
x = np.arange(1, 10).reshape((3, 3))
print np.linalg.norm(x, 'fro')
print np.linalg.cond(x, 'fro')
Q11. Calculate the determinant of x.
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.
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.
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
Q14. Return the sum along the diagonal of x.
x = np.eye(4)
out1 = np.trace(x)
out2 = x.diagonal().sum()
assert out1 == out2
print out1
Solving Equations and Inverting Matrices¶
Solutions using np.linalg.inv() to compute matrix inverses, verified by checking that the product of a matrix and its inverse equals the identity matrix.
Q15. Compute the inverse of x.
x = np.array([[1., 2.], [3., 4.]])
out1 = np.linalg.inv(x)
assert np.allclose(np.dot(x, out1), np.eye(2))
print out1