How to calculate Eigenvalues/Eigenvectors

Course Q&A Math for Machine Learning Linear Algebra for Machine Learning and Data Sc...
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Hello! I have found the course to be very well explained so far, and I’m able to grasp everything with many re-watches and time to let new concepts sink in, but I’m feeling very lost in knowing how to calculate eigenvalues/eigenvectors.

For example, this slide after the “quiz” in the “Eigenvalues and Eigenvectors” video gives the answers, but is missing the calculations on how to get there, that are provided in other lessons.

I’m feeling frustrated and upset, because I’m feeling very stuck even after days of working on this.

I tried watching the Serrano.Academy video, but the section that gets to the Computation just simply says that it was computed via Wolfram Alpha.

Can anyone please recommend any helpful resources? I haven’t taken a math course in over 15 years before this one, and a lot of what I’m finding online is explained in terms that I don’t understand.

Thank you in advance :slight_smile:

Slide that I was referring to:

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Have you seen some cursus of pre-calculus on coursera? Perhaps it can help.

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Hello @pinkninja!

The slide that you have shared has also given us a step-by-step instructions from the matrix to the eigen values. We start with the characteristic polynomial, and then solve for \lambda. You can actually play this video again, and watch from around 3:00 and it will guide you through the same set of steps to get to some eigenvalues.

As for the eigenvectors, my suggestion would be for you to watch the video from 4:00 because it also provided step-by-step guidance to the eigenvectors from eigenvalues. You only need to replace the matrix with the one from the quiz and work through the steps. Then you should be able to get the eigenvector answers on the slide that you have shared.

Cheers,
Raymond

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I am also struggling to understand the calculations. I get how to get the eigenvalues but not the eigenvectors. So far the course has been very easy for me but I struggle with this part. For example how do you calculate this?:

9x + 4y= 11x
4x + 3y= 11y

and
9x + 4y= x
4x + 3y= y

I haven’t seen expressions like this before ( without a whole number as dot product). I find it ridiculous that there is only one , the last, video on such an important topic such as eigenvectors.

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Combine like terms:

9x + 4y = 11x becomes 4y = 2x
4x + 3y = 11y becomes 4x = 8y

Both equations reduce to x = 2y, so [x,y] can be any scalar multiple of [2,1].

Likewise:
9x + 4y = x becomes 4y = -8x
4x + 3y = y becomes 4x = -2y

Both equations reduce to y = -2x, so [x,y] can be any scalar multiple of [1,-2].

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Thanks so much. Managed to figure out at the end and passed the quiz. Now on to the programming assignment

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Sorry for getting back late @Ivana_Petkova, but great job figuring it out yourself and there is nothing better than that! :wink:

Cheers,
Raymond

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Apologies for the delay- thanks a lot for your help!

I think part of my confusion, which I didn’t articulate is that I didn’t know what a characteristic polynomial was.

In case anyone reads this later and is curious, it can be calculated using the Quadratic equation, and this site was a good refresher

I also found this page quite helpful to understand why certain values (convenience values) were being used

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Thanks for this explanation. The video just seemed to skip past how to solve the equations after finding eigenvalues. Trying to search how to solve them also came up with complicated solutions. Yours is simple and straightforward. One question, where you have for example, x = 2y, how do we know that y is 1? and same with in y = -2x, how do we know that x is 1?

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I’ve just finished the course in Audit mode for now. But see this post Eigenvalues & Eigenvectors quiz- how was the eigenvector derived? - #6 by Elemento.

As with others the eigenvectors are confusing initially. I’ve studied this stuff before but it’s been decades since I’ve done any math!

To answer your question, there are infinitely many eigenvectors, since they can be scalar multiples of each other. So for x = 2y, how do we know that y is 1? We don’t. We’re just choosing the simplest integer values that satisfy the equation. We could have chosen y = 2, in which case x = 4. Similar reasoning for y = -2x. Let x = 1, then y must be -2. But we could have said x = -1, in which case y = 2.

Btw, at various points through the course I also consulted other material, e.g., 3Blue1Brown’s Essence of Linear Algebra series on YouTube. I often find it helps if you can approach material from different angles. I also have an old Math A-level book that covers the topic. So I referred to that as well.

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thanks bs80! agree the video goes way to fast over this part

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I am about to complete this course. While going through this course, I also watched the series of 3Blue1Brown on linear algebra. It has helped me a lot.
3Blue1Brown :- Linear Algebra

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