Linear Algebra.ipynb

Kernel: Python 3 (Ubuntu Linux)

Linear Algebra

We can perform basic linear algebra using numpy

import numpy as np

# Define matrix and vector
x = np.array([0,1,2])
A = np.array([[2,1,2],[3,2,0],[1,0,1]])

# matrix-vector product
y = A@x

# element-wise operations
x[0] = 1
A[0,1] = 2

# extracing rows and columns
print(A[1,:])
print(A[:,1])

# copy matrix
B = A.copy()

# transpose
B.T

[3 2 0]
[2 2 0]

array([[2, 3, 1],
       [2, 2, 0],
       [2, 0, 1]])

Exercise

What happens when you define a new matrix using C = A? Hint: try changing an element in the new matrix and inspect both C and A.

Matrix and vector norms

# vector p-norm and induced matrix norm
np.linalg.norm(x,2)
np.linalg.norm(x,1)
np.linalg.norm(x,np.inf)

np.linalg.norm(A,2)
np.linalg.norm(A,1)
np.linalg.norm(A,np.inf)

# Frobenius norm
np.linalg.norm(A,'fro')

5.196152422706632

Special arrays

# all ones or zeros
y = np.zeros(4)
F = np.ones((4,4))

# normally distributed random elements
r = np.random.randn(4)

# identity matrix
I = np.eye(3)

# banded matrix
L = np.diag([1,1],k=-1)

Exercise

Create a 10 x 10 matrix with $a_{ii} = -2$, $a_{i,i+1} = a_{i,i-1} = 1$

Eigenvalues and vectors

A = np.array([[2,1,2],[3,2,0],[1,0,1]])

l,V = np.linalg.eig(A)

print('eigenvalues of A:',l)
print('first eigenvector of A: ',V[:,0])

eigenvalues of A: [ 4.0861302  -0.51413693  1.42800673]
first eigenvector of A:  [-0.5613869  -0.80731332 -0.18190642]

Exercise

Compute the eigenvalues and vectors of $$A = \left(\begin{matrix} 3 & 1 \\ 0 & 3 \end{matrix}\right)$$ What is special about this matrix?

Sparse matrices

Large matrices that have few non-zero entries are more efficiently stored as sparse matrix:

# import scipy module for sparse matrices
import scipy.sparse as sp

# sparse identity matrix
D = sp.eye(100)

# random sparse matrix
R = sp.random(100,100,density=0.01)

# plot a sparse matrix
import matplotlib.pyplot as plt
plt.spy(R)

<matplotlib.lines.Line2D at 0x7f512272fba8>

Exercise

Check what the command diags([1, -2, 1], [-1, 0, 1], shape=(4, 4)) produces