Eigenvalue and EigenVector with Numpy

Image is from wikipedia

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that only changes by a scalar factor, when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and v is a vector in V that is not the zero vector, then v is an eigenvector of T if T(v) is a scalar multiple of v. This condition can be written as the equation

Definition:

let A be an n by n matrix.A scalar $\: \lambda \:$ is called an Eigenvalue of matrix A if there is a non zero vector $\: \bar{x} \:$ such that $\: A\bar{x} = \lambda \bar{x} \:$. Such a vector $\: \bar{x} \:$ is called an eigenvector of matrix A corresponding to eigenvector $\: \bar{x} \:$

Example

show that $\bar{x} = \left\lgroup \matrix{2 \cr 1} \right\rgroup \:$ is an eigenvector of \( : A = \left\lgroup \matrix{3 & 2 \cr 3 & -2} \right\rgroup : \) corresponding to $\: \lambda = 4 \:$

Solution

from $A\bar{x} = \lambda \bar{x} \:$ we want to show that \( \left\lgroup \matrix{3 & 2 \cr 3 &-2} \right\rgroup \) $\left\lgroup \matrix{ 2 \cr 1} \right\rgroup$ = $\left\lgroup \matrix{2 \cr 1} \right\rgroup \left\lgroup \matrix{ 4} \right\rgroup \:$ we simply multiply the matrix and if the both side are equal then we say the vector is an eigenvector corresponding to the scalar eigenvalue

solving the above matrixes give $\left\lgroup \matrix{8 \cr 4 } \right\rgroup = \left\lgroup \matrix{8 \cr 4} \right\rgroup \:$ Hence $\bar{x} = \left\lgroup \matrix{2 \cr 1} \right\rgroup \:$ is an Eigenvector of $A = \left\lgroup \matrix{3 & 2 \cr 3 & -2} \right\rgroup \:$ corresponding to an eigenvalue $\lambda = 4$

Solving Eigenvector and Eigenvalue of a matrix using Numpy

>>import numpy as np

>>A = np.array([[3,2],[3,-2]])

>>A
>>array([[ 3,  2],
     [ 3, -2]])

>>eigenvalue, eigenvector = np.linalg.eig(A)

>>eigenvalue
>>array([ 4., -3.])

>>eigenvector
>>array([[ 0.89442719, -0.31622777],
                   [ 0.4472136 ,  0.9486833 ]])

we usually get two eigenvalues which we will subsequently use to get the corresponding eigenvector of the given matrix

Mustapha Omotosho

constant learner,machine learning enthusiast,huge Barcelona fan

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