Linear algebra is a branch of mathematics concerned with vector spaces and the mappings between those spaces. NumPy has a package called linalg that supports powerful linear algebra functions. We can use these functions to find eigenvalues and eigenvectors or to perform singular value decomposition:

>>> A = np.array([[1, 4, 6],
    [5, 2, 2],
    [-1, 6, 8]])
>>> w, v = np.linalg.eig(A)
>>> w                           # eigenvalues
array([-0.111 + 1.5756j, -0.111 – 1.5756j, 11.222+0.j])
>>> v                           # eigenvector
array([[-0.0981 + 0.2726j, -0.0981 – 0.2726j, 0.5764+0.j],
    [0.7683+0.j, 0.7683-0.j, 0.4591+0.j],
    [-0.5656 – 0.0762j, -0.5656 + 0.00763j, 0.6759+0.j]])

The function is implemented using the geev Lapack routines that compute the eigenvalues and eigenvectors of general square matrices.

Another common problem is solving linear systems such as Ax = b with A as a matrix and x and b as vectors. The problem can be solved easily using the numpy.linalg.solve function:

>>> A = np.array([[1, 4, 6], [5, 2, 2], [-1, 6, 8]])
>>> b = np.array([[1], [2], [3]])
>>> x = np.linalg.solve(A, b)
>>> x
array([[-1.77635e-16], [2.5], [-1.5]])

The following table will summarise some commonly used functions in the numpy.linalg package:

>>> a = np.array([[1, 0],[0, 1]])
>>> b = np.array( [[4, 1],[2, 2]])
>>> np.dot(a,b)
array([[4, 1],[2, 2]])

>>> a = np.array([1, 1, 1])
>>> b = np.array([3, 5, 1])
>>> np.inner(a,b)
9

>>> a = np.arange(3)
>>> np.linalg.norm(a)
2.23606

>>> a = np.array([[1,2],[3,4]])
>>> np.linalg.det(a)
-2.0

>>> a = np.array([[1,2],[3,4]])
>>> np.linalg.inv(a)
array([[-2., 1.],[1.5, -0.5]])

>>> a = np.array([[1,2],[3,4]])
>>> np.linalg.qr(a)
(array([[0.316, 0.948], [0.948, 0.316]]), array([[ 3.162, 4.427], [ 0., 0.632]]))

>>> a = np.array([[1,3],[2,4]])
>>> np.linalg.cond(a)
14.933034

>>> np.trace(np.arange(6)).
reshape(2,3))
4