Eigenvalues and Eigenvectors play a critical role in determining observable measurements of quantum systems, in evaluating the stability of mechanical structures, in analyzing the feedback loops of electric circuits, and in many other areas. To fully understand physics and engineering, it is necessary to understand Eigenvalues and Eigenvectors. Suppose we have an object. Here we are viewing the object from different perspectives. Now, suppose we apply what we call a linear transformation. As we again view the object from different perspectives, we see that the object is now distorted. Let us consider the points on this object before the transformation. Each of these points can be represented by an arrow. When the transformation is applied, the length and direction of each arrow can change. There are some arrows that point along the same line both before and after the transformation. For example, these three arrows will simply have their lengths multiplied by a constant. The length of the orange arrow is multiplied by negative one. The length of the blue arrow is multiplied by two. And the length of the green arrow is multiplied by one half. We say that negative one, two, and one half are the “eigenvalues” of this transformation. The arrows that point along the same line both before and after the transformation are what we call the “eigenvectors.” The amounts by which the lengths of each of these arrows is multiplied are what we call the eigenvalues. The eigenvalues and eigenvectors dictate the nature of the transformation for the entire object. Consider, for example, the point symbolized by this white arrow. The white arrow is not aligned with the directions of any of the eigenvectors, which are symbolized by the orange, green, and blue lines. But, the white arrow can be thought of as a combination of arrows that are parallel to the directions of these three lines. When the transformation is applied, the lengths of each of these arrows is multiplied by the eigenvalue associated with that direction. That is, the orange, green, and blue arrows continue pointing in the same direction, but are multiplied by “-1”, “1/2”, and “2.” In this way, the transformation of any point on this object can be determined by using the eigenvalues and eigenvectors. The eigenvectors do not necessarily have to be 90 degrees to one another. For example, let us consider this new object, and this new transformation. The red and green arrows are eigenvectors of this transformation. The green arrow is multiplied by one, and the red arrow is multiplied by two. Now consider the transformation of this white arrow. The white arrow can be thought of as the combination of arrows parallel to the green and red lines. During the transformation, this green arrow is multiplied by one, and this red arrow is multiplied by two. The eigenvalues and eigenvectors do not necessarily have to consist of real numbers. The eigenvalues of this transformation are the imaginary numbers “i” and negative “i.” And the eigenvectors are the following. Let us say that the red arrow is signified by a “1” followed by a “0.” And let’s say that the yellow arrow is signified by a “zero” followed by a “one.” Before the transformation, the blue arrow is in the same direction as the red arrow, but double its length. Therefore, using the notation described, the blue arrow before the transformation would be signified by a “2” followed by a “0.” After the transformation, the blue arrow is pointed in the opposite direction of the yellow arrow, and double its length. Therefore, the blue arrow after the transformation would be signified by a “0” followed by a “-2.” Let us consider the blue arrow before the transformation. The blue arrow can be thought of as the sum of the two eigenvectors. Suppose we multiply each eigenvector by its corresponding eigenvalue. The result is the new direction of our blue arrow after the transformation. Therefore, everything discussed here is still true even for imaginary values for eigenvalues and eigenvectors. Much more information about linear transformations is available in the video “Linear Algebra – Matrix Transformations.” Please subscribe for notifications when new videos are ready.

Great explanation!

Great video as always. Great choice of music, also as always!

Very informative video. Easy to comprehend and with the bonus of a classy music. It's like the unification of science and arts..

your videos are so helpfull.thanks.

Gilbert Strang Rocks!

Love the Hungarian rhaspody no. 2 in the back making it evermore dramatic.

Love the physics representations too.

How do people even get a physics degree without knowing about this. It's basic math bois

thank you soo much

Nice explanation.

it is more than beautiful.

Amazing….. Extra ordinary….. You did a great job……. And we are so helpfull because of you….. I have never thought it like that… I love u….. ❤︎😍

good video

U are awesome ,the music behind and the voice is so pleasent which makes me concentrate more !!!thanks

The last proof was Great!

One of the best channel, i'm just wondering,thanks to you and your teammates to help us to gain knowledge

Please made and video about the relationship between mass increasing with the speed of light

Best video on the internet for eigenvectors

Hi Eugene , your videos are super intuitive and magnificent. keep up the great work..

Omg ..this video is amazing😊

Very good explanation i n 3D way…Thanks a lot……It make me understand in easy way….

I loce these colourfull demontrations describes thing better and you know wich is wich its great for following.

What topic of maths does curl come from?

How you make these videos

Thanks for the video. I can understand these concepts but What is the purpose of them? Please add the applications of them.

awsome

everything made reason until you started talking imaginary numbers. Because now it seems to me that if we have a linear transformation in a two dimensional plane EVERY vector is an eigenvector. It just takes to find a suitable complex number as eigenvalue. It will give you the rotation and the length of the vector in the new transformation.

Thank you for this good video

I was hoping you would explain what complex eigenvector and eigenvalue mean. Real eigenvector and eigenvalues signify directions in which the applied linear transformation do not distort. But how what "direction" do complex eigenvectors point to?

"eigenvalues and eigenvectors play a critical role in determining observable measurements of quantum systems in evaluating the stability of mechanical structures in analyzing the feedback loops of electric circuits and in many other areas".

I only found this to be informative and the rest is absurd.

I don't understand why do you need to transform the object in the first place? Do it need to be linear? If yes, why it must be linear? What kind of object that you can't transform? Why do you need to assume the points and the arrows? How do you assume it correctly? Why it must be aligned? What is the physical meaning the object in reality? It's absolutely abstract and indoctrinative.

Before it explained, it jumps into real and imaginary number. Why do I need it if I haven't understand anything before?

And then it ends with "much more information about linear transformations is available in the video linear algebra matrix transformations". Not indicating any sequel whatsoever.

I'm disappointed, I expect more. Please make more.

This was so helpful, thank you

practical example of how to actually use it along with this would be perfect and thanks for creating such useful videos and making science much easier for us.

This is helping me understand System-Dynamics and Control Feed back in my ME class. Thank you.

please make the video on logerathemic

Does anyone know what music is played in the background?

Yet another excellent video. Thanks

<3 <3 <3 <3

Russians are always brilliant…

Thanks so much, your video helped in understanding the basic concepts.

Looking forward to see more content from you.

Eugene! I did not even realize you made a video on this topic until YouTube recommended me this video a few hours after struggling to find clarification on the internet. My god, there is a lack of visual explanations on the net that explain eigenvectors and eigenvalues geometrically so thank you so much. :') You've once again freed my from my mental anguish. Thank you!

I have no idea of what this is….

OMG! All the way I was thinking what would be the next sentence 🙂

You guys relief me from a frustration scenario. Thank you. Keep it up.

Very cool and understandable video, but sometimes the pacing is a bit strange.

If I understand properly there are only certain vectors that do not change direction during a transformation. So how would you know which ones are these eigenvectors? Are they not pretty much having to do with the applied transformation? If you knew those you could just transform the points with the original transformation?

If you still don't understand you can watch 3Blue1Brown's playlist on Linear Transformation.

Leibniz said why is there existance of Universe ???

Significantly important for describing Energy in particle space

So Elegant! mathematics + Liszt&Bach

Beautiful!! Thankyou!!

Are you GOD…atleast you seem to be one.

Excellent👍👏😆

I love your videos

To be able to get a base of eigenvectors the linear transformations matrix has to be diagonizable though.

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Thanks.

A comment a comment!

visualization wise its good just background music could have been lower.

This is how Mathematics should be studied💕💕💕

do you make this cool 3d object using blender?

The quality of videos on this channel is just WOW ! Please don't stop making them !

Thank you so much for such wonderful explanations!

Physics

+

Franz Liszt

=

Ecstasy

This video makes me happy. I was worried about the meaning of the eigenvalues and the eigenvector. It's a long way to go but these videos make the trip happier

Make more videooo

Thanks u so much

Can u explain in hindi. What is eigen value and eigen vector

It's amazing!!!!!!

Your work is just so good!

Love u sooooooooooooooooooooooooooooo much.thnx a lot.opened my eye

commenting so that search engine recommends this first

Liked @3:43

Brilliant man. Really knows how to explain.

These videos are great except for the pretentious, annoying music! Why have it? It adds nothing to the content but distraction and irritation, ruining what would otherwise be interesting. Bad choice.

Here come Liszt again 🙂

Por qué tratar de solucionar la ecuacion de onda se convierte en un problema de autovalores de sturm lioville regular?

Life is a learning game play it be confident

Thanks a lot. The video is really helpful.

This is AMAZING, thank you so much and keep up the great work!

nice

I finally understood eigenvector.. thank you so much:)

What the white line that's parallel to the eigenvector called as?

I only knew how to find the engenvalues but never understood their true meaning.

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Pregunta no hablo inglés pero eso es una neurona comportamiento vivo

La representación de una neurona que nos han enseñado es como un transistor con tres patas esta representación es de una neurona qué tiene un comportamiento vivo multilateral en tres dimensiones que no significa que la neurona tiene conexiones fijas sino que fabrica sus conexiones en base a su pensamiento lógica nuestro pensamiento lógica las neuronas también son inteligentes saben Qué camino tomar

I suspect that the 'machines' are trying to teach us new things

Great job dear!

This is great!!

Thank you so much! I never thought Eigen vectors could be this simple to understand.

Wow, our universe is even more dualistic than I thought!

Sooooooo just to make it clear. Any vector after a linear transformation can be described as a linear combination of eigenvectors. An eigenvector is a vector which after the linear transformation doesn't change its direction. However it changes its magnitude and way given by the eigenvalue. Is that right??

I had always wondered whether all these mathematical concepts had a physical meaning and you just showed it so beautifully. Thank you

You are awesome

Please repost without the music playing. It is very distracting. Thank you for your work.

Very good video as always.

Some of you viewers might want to go check the linear algebra series by 3blue1brown.

Both of your channel do a great job of actually “showing” maths and that is so insightful.

Keep up the good work!

not your best video. It was 10minutes too long for the points you kept repeating and the geometry was too cluttered in the first example that it would have been so much better to summarize everything with just a cube instead.

👍

Excellent.

thanks for the demonstration:)

I've seen this with my own eyes.

I hope this channel doesn’t mind that I use these videos for inspiration for the posters I draw up all over my apartment.

I’ve been trying to understand the eigen cousins for FOUR decades. This was the mist coherent ever.

I’ve been playing go even longer and have about the same level of comprehension.

I get completely lost in thoughts when i watch your videos..thank you♥️

But that wide variety of music selection!🙄🙄