Standard linear algebra operations

Matrix multiplication

Two arrays can be manipulated in the usual linear-algebra way using numpy.matrxmultiply. Here is an example:

>>> import numpy as N
>>> import numpy.random
>>> A = numpy.random.rand(5, 5)
>>> x = numpy.random.rand(5)
>>> b = N.dot(A, x)
>>> A
array([[ 0.78447957,  0.38316272,  0.31135453,  0.70738504,  0.07551859],
       [ 0.43140155,  0.55929425,  0.28448864,  0.38397626,  0.14849022],
       [ 0.88869344,  0.93048982,  0.70973804,  0.04341364,  0.27067106],
       [ 0.95290594,  0.17021306,  0.98377969,  0.73476449,  0.40728296],
       [ 0.05918968,  0.357716  ,  0.33474359,  0.99625301,  0.68345456]])
>>> x
array([ 0.47874927,  0.52254844,  0.53053969,  0.07469784,  0.67971556])
>>> b
array([ 0.84514734,  0.77933734,  1.47545378,  1.39880365,  0.93182836])

Solving systems of linear equations

To solve a system of equations Ax=b that is given in matrix form, we can use the linear algebra package of numpy:

>>> import numpy.linalg as LA
>>> x = LA.solve(A,b)
>>> x
array([ 0.47874927,  0.52254844,  0.53053969,  0.07469784,  0.67971556])

Computing Eigenvectors and Eigenvalues

Here is a small example that computes the Eigenvectors and Eigenvalues (eig) of the unity matrix.

>>> import numpy
>>> import numpy.linalg as LA
>>> A = numpy.eye(3)
>>> print A
[[ 1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0.  1.]]
>>> evalues, evectors = LA.eig(A)
>>> print(evalues)
[ 1.  1.  1.]
>>> print(evectors)
[[ 1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0.  1.]]

 Curve fitting of polynomials

Here is a small example that computes the [trivial] Eigenvectors and Eigenvalues (eig) of the unity matrix (eye)):

>>> import numpy
>>> xdata = numpy.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0])
>>> ydata = numpy.array([0.0, 0.8, 0.9, 0.1, -0.8, -1.0])
>>> z = numpy.polyfit(xdata, ydata, 3)
>>> print z
[ 0.08703704 -0.81349206  1.69312169 -0.03968254]
>>> p = numpy.poly1d(z)
>>> xs = [0.1 * i for i in range(50)]
>>> ys = [p(x) for x in xs]
>>> import pylab
>>> pylab.plot(xdata, ydata, 'o', label = 'data')
[]
>>> pylab.plot(xs, ys, label='fitted curve')
[]
>>> pylab.ylabel('y')

>>> pylab.ylabel('x')

>>> pylab.savefig('polyfit.pdf')
>>> pylab.show()