Matrix Operations using Python

Dr. Hari V S
Department of Electronics and Communication
College of Engineering Chengannur

What You will Learn

• You will learn to create matrices
• You will learn to solve linear equations by computing the inverse of a matrix.
• You will learn to compute the eigen values of a matrix

Matrices using Python

A $5\times 3$ matrix is created, printed and displayed as an image using the code below. Its transpose is generated by the $transpose()$ function.
from scipy import *
from pylab import *
x=mat([[1,2,3,4,5],[4,5,6,9,15],[7,8,9,23,25]])
print x
print shape(x)
print transpose(x)
matshow(x)
show()

Solution of Linear Equations

Linear systems are represented by matrix equations. See the electrical network in the figure below. All resistances are in $\Omega$. The network is described by the set of equations. \begin{equation} 9I_{1}-4 I_{2}=20 \end{equation} \begin{equation} -4I_{1}-6 I_{2}=-10 \end{equation} \begin{equation} 8I_{3}-3 I_{4}=-20 \end{equation} \begin{equation} -3I_{3}+7 I_{4}=10 \end{equation}

Or as a matrix-vector product as \begin{equation} \begin{bmatrix} \phantom{-}9&-4&\phantom{-}0&\phantom{-}0\\ -4&\phantom{-}6&\phantom{-}0&\phantom{-}0\\ \phantom{-}0&\phantom{-}0&\phantom{-}8&-3\\ \phantom{-}0&\phantom{-}0&-3&\phantom{-}7\\ \end{bmatrix} \begin{bmatrix} I_{1}\\ I_{2}\\ I_{3}\\ I_{4}\\ \end{bmatrix} = \begin{bmatrix} \phantom{-}20\\ -10\\ -20\\ \phantom{-}10\\ \end{bmatrix} \end{equation} We will get the solution for currents if we pre-multiply this equation with inverse of the resistance matrix. The following yields the solution for currents.
from scipy import *
import scipy.linalg as lin
rmat=mat([[9,-4,0,0],[-4,6,0,0],[0,0,8,-3],[0,0,-3,7]])
sourcevect=mat([20,-10,-20,10])
Ivect=lin.inv(rmat)*transpose(sourcevect)
print Ivect
When run the code prints the current values as $I_{1}=2.10526316$, $I_{2}=-0.26315789$, $I_{3}=-2.34042553$ and $I_{4}=0.42553191$.

Eigen Values of a Matrix

Let us look at the matrix equation \begin{equation} [\mathbf{A}][\mathbf{X}]=[\mathbf{b}] \end{equation} where $\mathbf{A}$ is an $N\times$ matrix and $\mathbf{X}$ and $\mathbf{b}$ are $N\times 1$ vectors. It is not necessary that the vectors $\mathbf{X}$ and $\mathbf{b}$ need not be in the same direction. Let us look into this with $N=2$. Let $\mathbf{A}=\begin{bmatrix} 1 & 2\\3&4\\ \end{bmatrix}$ and the vector $\mathbf{X}=\begin{bmatrix}\phantom{-}1\\ -1\\ \end{bmatrix}$ making the vector $\mathbf{b}=\begin{bmatrix}-1\\ -1\\ \end{bmatrix}$. the two vectors $\mathbf{X}$ and $\mathbf{b}$ are plotted along the orthogonal coordinates $x_1$ and $x_2$ as shown in the figure below. You can see that they are in different directions.

For $\mathbf{b}$ or $\mathbf{A}\mathbf{X}$ to be in the direction of $\mathbf{X}$ \begin{equation} [\mathbf{A}][\mathbf{X}]=\lambda [\mathbf{X}] \end{equation} where $\lambda$ is a constant. Transposing this equation, gets \begin{equation} [\mathbf{A}-\lambda \mathbf{I}][\mathbf{X}]=[\mathbf{0}] \end{equation} $\mathbf{X}=\mathbf{0}$ is a trivial solution which makes \begin{equation} [\mathbf{A}-\lambda \mathbf{I}]=[\mathbf{0}] \end{equation} or \begin{equation} |\mathbf{A}-\lambda \mathbf{I}|=0 \end{equation} where $\mathbf{I}$ is the $N\times 1$ identity matrix. The above eaquation is called the characterstic equation and the solutions for $\lambda$ are called eigen values of the matrix $\mathbf{A}$ . The vectors $\lambda \mathbf{X}$ are called eigenvectors. See the red colored vector in the above figure. Eigen values and vectors are useful in trend analysis and reduction of dimensionality of data. the eigen values of the above resistance matrix is computed with the following code.
from scipy import * import numpy.linalg as lin
rmat=mat([[9,-4,0,0],[-4,6,0,0],[0,0,8,-3],[0,0,-3,7]])
eig_vals=lin.eig(rmat)
print eig_vals[0]
print eig_vals[1]
print "The rank of the matrix is " + repr(lin.matrix_rank(rmat))
The eig module in linalg computes the eigen values and normalized eigen vectors and returns as two arrays. eig_vals[0] contains the eigen values as $[ 11.77200187+0.j 3.22799813+0.j 10.54138127+0.j 4.45861873+0.j]$. eig_vals[1] contains the corresponding normalized eigen vectors. Here they are $[ 0.82192562 0.56959484 0. 0. ]$, $[-0.56959484 0.82192562 0. 0. ]$, $ [ 0. 0. 0.76301998 0.6463749 ]$ and $ [ 0. 0. -0.6463749 0.76301998]$ respectively. The matrix_rank module returns the rank of the matrix.

What You Learned

• You learned to create matrices
• You solved linear equations by computing the inverse of a matrix.
• You computed the eigen values and rank of a matrix