NumPy - linear algebra

Linear algebra is a branch of mathematics concerned with vector spaces and the mappings between those spaces. NumPy has a package called linalg. This page is meant only to familiarize you with the NumPy’s linear algebra functions for those who are interested.

A \(1 \times N\) dimensional vector \(x\)

\[\begin{split}x = \begin{pmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{N} \end{pmatrix}\end{split}\]

and its transpose \(\mathbf{x}^{T} = (x_{1}, x_{2},\ldots,x_{N})\) can be expressed in python as

>>> import numpy as np
>>> x = np.array([[1,2,3]]).T
>>> xt = x.T
>>> x.shape
(3, 1)
>>> xt.shape
(1, 3)

A column matrix in NumPy.

\[\begin{split}x = \begin{pmatrix} 3 \\ 4 \\ 5 \\ 6 \end{pmatrix}\end{split}\]

>>> x = np.array([[3,4,5,6]]).T

A row matrix in NumPy.

\[\begin{split}x = \begin{pmatrix} 3 & 4 & 5 & 6 \end{pmatrix}\end{split}\]

>>> x = np.array([[3,4,5,6]])

General matrices you are already familiar with.

\[\begin{split} A_{m,n} = \begin{pmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{pmatrix}\end{split}\]

Common tasks

Matrix determinant

>>> a = np.array([[3,-9],[2,5]])
>>> np.linalg.det(a)
33.000000000000014

Matrix inverse

>>> A = np.array([[-4,-2],[5,5]])
>>> A
array([[-4, -2],
       [ 5,  5]])
>>> invA = np.linalg.inv(A)
>>> invA
array([[-0.5, -0.2],
       [ 0.5,  0.4]])

>>> np.round(np.dot(A,invA))
array([[ 1.,  0.],
       [ 0.,  1.]])

Because \(AA^{-1} = A^{-1}A = I\).

Eigenvalues and Eigenvectors

>>> a = np.diag((1, 2, 3))
>>> a
array([[1, 0, 0],
       [0, 2, 0],
       [0, 0, 3]])
>>> w,v = np.linalg.eig(a)
>>> w;v
array([ 1.,  2.,  3.])
array([[ 1.,  0.,  0.],
       [ 0.,  1.,  0.],
       [ 0.,  0.,  1.]])

This is by no means a complete list—also the SciPy package has additional functions if this is an area of interest.

Bibliographic notes

1. Duda, R. O., Hart, P. E. & Stork, D. G. Pattern Classification, John Wiley & Sons, Inc., 2001.