Eigen Vector and Eigen Values

Eigen vector is the direction in a coordinate space defined by a metrics which doesn’t change its direction with metrics transformation.

Eigen value is a scaler number which is multiplied with Eigen vector to give same result as Eigen vector multiplier with existing metrics.

Lets suppose A is metrics and v is a Eigen vector then 

Av = λv , is the representation of Eigen vector ‘v’ with Eigen value ‘λ’.

Let us suppose metrics A and vector v as:-

Now make a dot product of A and vector v as k1 = A.v and calculate the slope of direction represented by output

Repeat the process for k2,k3,k4,k5…. 

We can say that after repeated dot product calculation the slope is converging towards a certain value i.e. direction is getting constant and only magnitude is getting changed. It is what a Eigen vector do by defining a direction which will not change by matrix transformation or any scaler multiplication.

Let’s take one more example

Here we will consider matrix A and vector B as mentioned below.

When we calculate dot product between A and B the output turned out to be vector [3,6] which is not in the direction of vector B i.e. [1,1]. So B will not be considered as Eigenvector.

Let’s consider same vector A and do dot multiplication using B = [1,2].

Here we can see the product turn out to be a vector [5,10]. which is just the extension in the direction of vector [1,2] i.e. 5 *[1 , 2]. So if B = [1 , 2] for vector A then B will be considered as eigenvector for A and the multiple 5 will be considered as eigenvalue of the matrix A.

Let’s calculate the Eigen vector for same case….

We will start from the equation which should be true. i.e.

Av = λv

We can put an identity matrix in RHS as multiplying identity matrix with any existing matrix doesn’t changes the output.

Av = λIv

Bring everything on left side will results into

Av – λIv = 0

det(A – λI) = 0

Now we will solve our problem using the equation formed: –

Since, it is a quadratic equation it will give two value of λ as Eigen values on solving as: –

λ1 , λ2 = {2.36 , 0.23}

Now, We will find Eigen vectors by solving Av = λv

Now, the calculated value will be converted into unit vector i.e. ||v|| = 0, We can divide the calculated v1 vector by euclidean value to get our first Eigen vector as 

and similarly second Eigen vector can be calculated by using 0.23 as Eigen value.

Slope of this Eigen vector will be 0.41/0.91 = 0.45, it is the same value where our vectors were converging earlier after continuous multiplication of vector [-1 , 1].