Visualizing Diagonalization & Eigenbases

Diagonal transformations are really nice to visualize geometrically. In 2D they are just a combination of horizontal and vertical stretching.

While a generic matrix isn't quite this nice, if you can find a basis of eigenvectors, then the transformation "looks" like stretching and compressing along those eigenvectors by the values of the eigenvectors. This makes it pretty nice, but we can do better.

When we diagonalize a matrix, this is a composition of transformations. You first apply a change-of-basis to convert from the standard basis to the eigenbasis. Then you apply the nice diagonal transformation. Finally, you convert back.

**************************************************
Now it's your turn:
1) Summarize the big idea of this video in your own words
2) Write down anything you are unsure about to think about later
3) What questions for the future do you have? Where are we going with this content?
4) Can you come up with your own sample test problem on this material? Solve it!

Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master math means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.

***************************************************
Want more ideas for learning math effectively?
►How to Watch Math Videos:

****************************************************
►Want some cool math? Check out my "Cool Math" Series:

****************************************************
Course Playlists:

*****************************************************
*****************************************************

This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.

BECOME A MEMBER:

MATH BOOKS & MERCH I LOVE:

[richardojuan]

4:43 there is a type at the right top corner: the eigenvector should be (-1, 1) instead of (1, 1)

[alkankondo]

Absolutely, absolutely excellent! The visuals; the clear, thorough explanations; the excellent balance of example and theory: they all came together to make a very informative video that increased my intuition of Linear Algebra. Thank you for this video! I'm so glad I watched this.

[godknifetube]

Thank you so much! The visualisation not only makes the concept clear but also helps in the intuition. Brilliant!

[logandarby]

Heyo! Just wanted to tell you that you've absolutely saved my life. I was really struggling with the connection between diagonalization and eigenvalues, and this video turned it from a needlessly theoretical concept, into feeling almost obvious intuitively. Thanks a bunch!

[EilafBadr--]

Thank you for making linear algebra more Fun and visualizable

[sdsa]

i think this is important for PCA (Principle Component Analysis ) and i am super thankful for getting a intuitive visualization on this topic instead of having to memorize equations and proofs.. THANK YOU!

[supratimhalder]

Damn man! You are really underrated. You should have way more subscribers.

[MatthewFoulk]

The visuals at the end were just what I needed. I'd been struggling to understand why it was P^-1 that converted from the standard basis instead of P, the visuals helped me actually see what was happening. All around excellent video! Also this is the second time today that I stumbled across this channel (the first was related to Calc 3 and vector fields)

[matiassantacruz]

Holy cow you and your videos are amazing….in in uni know and discovered your channel because of Linear Algebra, and I have to say your content is wonderfully insightful and just EASY TO UNDERSTAND! Your online book is also amazing, you should be proud because you are carrying future generations of physicists, mathematicians and engineers. If there is anywhere we could make a donation, I’d be happy 😊. THANK YOU ♥️

[gabrielpereiramendes]

Thanks for the content.
I think this way, I can understand much more because I can see how the eigenvectors and eigenvalues are
important to linear transformation.
Visualizing the process turn it easier to learn.

[navjotsingh]

You are right up there with 3b1b in regards to excellent math content. Keep up the good work 😃 this is the time for mathematicians to shine and make videos explains all topics and building intuition. I really love this blow up of videos on YouTube, and I am subscribing to lots of math channels. I am truly forever grateful for your work. I say this with all my heart, thank you.

[GustavoMontanha]

clear and beautiful. my brain still refuses to see the mechanics behind all this, keeps taking me back to standard canonical brain.

[person]

Thank you so much for this fantastic video! Your videos are incredible, I've been binging them on and off the last few months.

[peanutsee]

Thank you Dr. Trefor, you are a genius.

[martianlightning]

WOW this was so helpful and exactly what i needed!! i somehow made it through two whole linear algebra classes without actually understanding the meaning of PDP^(-1) decomposition, struggling with the computations because i didn't really understand what i was doing. this video truly was a lightbulb moment for me, thank you so much!

[lennyatomz]

Bravo, Dr. Bazett!! This is a fantastic video and you have presented the materials elegantly. Thank you so much for your contributions to the world of Mathematics, sir! :-)

[fordtimelord]

The visuals in the last couple minutes are a nice touch.

[aidosmaulsharif]

man i wish you had more views, just sad, because the video is great. Professor Strang and his mit lectures are cool but they lack visualizations. Thank you very much

[eyadshami]

Thank you for the visual comparison between the two basis.

[cesarmorenoy]

You are the gold standard for education content