Finding eigenvectors and eigenvalues. Step-by-step.

I hope every data scientists knows how eigenvectors and eigenvalues are important in ML/DS area. But this post is rather for people who start studying data science from scratch and I would like to put my contribution just to help someone who seeks a simple answer. I won’t dwell on theory because I there are a lot of materials about the topic we will discuss in web, just want to focus on example.

The equation of a linear transformation:

Image for post
Image for post

Let’s look at the example a bit closer. Supposed, we have a matrix A:

Image for post
Image for post

That means that there is eigenvalues and eigenvectors that satisfy such equation:

Image for post
Image for post

If we apply matrix multiplication and draw up system of equations it will result in the following expression:

Image for post
Image for post

Apparently, we can express matrix as a system of equatioins for reducing complexity:

Image for post
Image for post
Image for post
Image for post
Image for post
Image for post
Source equation

Of course, you can omit these steps and write matrix immediately but that was important to know why we have lambda in diagonal. Let’s write a matrix, but be careful with x and y coefficients. Vector v cannot be zero. That means, x and y cannot have a trivial solution when x = 0 and y = 0. In this case the equations are lineary dependent (we will see it later) and a matrix determinant is equal to zero.

Image for post
Image for post

That is called characteristic equation of the matrix A. Let’s find the determinant and hence find the eignent values from equation:

Image for post
Image for post

Substitute these values to our equation (see Source equation above)

Image for post
Image for post
Image for post
Image for post

If there is no linear dependency (in our case x=y=0), then there in a mistakes in calculations. If we place any x for y (or vise versa) it will result in any infinite number of eigenvectors. That is why we can choose any y. Example suggests to use y=-1 just for simplicity.

Image for post
Image for post

Do the same for another lambda (y=-2)

Image for post
Image for post

We have found eigenvectors and eigenvalues.

Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store