What Does It Mean For a Matrix to be POSITIVE? The Practical Guide to Semidefinite Programming(1/4)

Video series on the wonderful field of Semidefinite Programming and its applications. In this first part, we explore the question of how we can generalize the notion of positivity to matrices.
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Timestamps:
0:00 Intro
0:41 Questions
2:50 Definition
6:09 PSD vs eigenvalues
7:40 (Visual) examples
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Credit:
🐍 Manim and Python : github.com/3b1b/manim
🐵 Blender3D: www.blender.org/
🗒️ Emacs: www.gnu.org/software/emacs/
Song: Thinking Ahead
This video would not have been possible without the help of Gökçe Dayanıklı.

[@hclimente]

Thank you for making this, finally I get a concrete, visual intuition of what positive definiteness means!

[@VisuallyExplained]

You are very welcome. Thanks for the nice comment!

[@light_rays]

The animations are so beautiful! And the geometric intuition was new to me, I like how it also gives a picture for positive linear functionals.

[@VisuallyExplained]

Thank you so much for the very nice comment, stay tuned for more!

[@tiga7659]

Amazing video! After years of studying at uni I finally understood the real intuition behind Positive definite matrices. Thank you so much! :D

[@MayankGoel447]

This is the best video out there to explain Positive Semidefinite Matrices! I finally understood what Positive Definiteness means visually after struggling with the topic for a long time.

[@piyukr]

Finally I got a geometric intuition of what positive definiteness means. Very well explained! Thank you so much.

[@moopoo123]

This is a BEAUTIFUL video. I am floored by your animations and how you communicated the relevance of PSD matrices.

[@krishnasinghrajput4206]

How and why is this channel so underrated. The quality of work is one of the finest.

[@abdulazizalhaidari7665]

What a great effect your videos have on the course of understanding of many people,

[@mspy2989]

Lol you literally uploaded this video the day before my convex optimization exam. Thanks!

[@VisuallyExplained]

Awesome!!!! Good luck for your exam!

[@anjishnu8643]

Thanks a lot for uploading this. I had been searching for a visual explanation on the topic for a long time. Also, really well explained.

[@VisuallyExplained]

Thank you very much. :)

[@praticandomatematica246]

Such a great explanation! I have never read a more clear definition of positive definite matrix.

[@bitflop]

Such beautiful and clear explanations. You are a legend in the making!

[@VisuallyExplained]

Thank you Ario!!

[@tylerperini]

This is an amazing series, and I look forward to sharing this with my students!

[@VisuallyExplained]

Please do!

[@vasanthakumarg4538]

Explanation and animations are really great. keep up the good work.

[@michaelho5138]

This is a really great video-love the vibe from the music and timing at the end 😂

[@user-ey1dg4xn2u]

Please keep making videos. So helpful and intuitive!

[@VisuallyExplained]

Thank you! Will do!

[@lithiumdeuteride]

Excellent video. Clear, concise, and illuminating.

[@VisuallyExplained]

Much appreciated!!!

[@amirreza08]

Just WOW! What a great explanation.Thank you!

[@jonasulmen5332]

huge fan!

[@VisuallyExplained]

Hooray! :)

[@pauledam2174]

Boom! What a great way to start out -with a single element matrix were the element is a positive number. Fantastic!

[@CHUAN-CHI]

Absolutely wonderful.

[@fadi.almasalmah]

what a wonderful explanation! keep going!

[@danielx1912]

Amazing amazing videos, cannot say how much I appreciate this!

[@VisuallyExplained]

Thank you!

[@ClearMath1]

My God, you've done an amazing job! I Hope you'll continue to grow, we Need more channels like this to Give credit to this beautiful topics with so smooth animations!

[@VisuallyExplained]

Thank you so much 😀 I checked your channel and you are doing such fine work yourself!

[@potatooflife8603]

this is amazing. thank you.

[@georgesms]

WOW! THAT WAS AWESOME!

[@andres_camarillo]

Amazing video. Thanks!

[@arcmchair_roboticist]

Wow this is an amazing video and a great channel!

[@VisuallyExplained]

Thank you so much!!

[@MaiNguyen-yp1sj]

wow, best explanation on positive definite that I can find. Thank you so much

[@VisuallyExplained]

You're very welcome!

[@bilenkeziban6237]

Great video, thank you!

[@MohammadRamadan-o5c]

Brilliant explanation!

[@jayantnema9610]

totally love your work dude!

[@priyankajain1691]

Nicely explained. Thank you

[@pp-xc8kb]

I cannot say anything more than amazing video! Thank you for your huge contribution!

[@VisuallyExplained]

Thanks! Glad this was helpful!

[@davidsal8308]

Awesome video!

[@taotaotan5671]

Awesome. I only encountered this concept from the error message when I am trying to generate some multivariate Gaussian random number with a specified covariance matrix. Nice to see the formal definition!

[@hansenmarc]

I ran across positive semidefinite matrices while studying machine learning, but never had a good intuition about them. Thank you for providing some conceptual insight into them.

[@ocamlmail]

Tremendously cool explanation, thank you very much!

[@VisuallyExplained]

You're very welcome!

[@fishinker]

So good, thank you

[@accountname1047]

This is a fantastic video

[@VisuallyExplained]

This is a fantastic comment!

[@ZhanCaitao]

Nice video!

[@sebastianmezaurotadze2581]

awesome video

[@giominor88]

This is amazing. I would love to see some "visual" explanation of SDP relaxations a la Lassere.

[@user-yk2zu5px3u]

thank you so so so much; ive been using the definition of positive definite for so many courses but I never understood why/how it is used i just merely did the calculation it's almost like a eureka for me thank you!!!!

[@VisuallyExplained]

Great to hear!!

[@stephaniedsouza3042]

I am sleepy and tired but this was still an awesome watch

[@jkgan4952]

Great Video! Thank you so much :))))))))))))))))))

[@igalleikin5040]

Amazing

[@josecarlosferreira4942]

Quadratic functions f(x) = x'Qx, (x is a vector and Q is a matrix), you can consider a symmetric matrix, because: x'Qx is a scalar, so x'Qx = (x'Qx)^T (transpose) x'Qx = x'Qx /2 + (x'Qx)^T / 2 = x'(Q+Q^T)x A=Q+Q^T A is symmetric because A=A^T, so f(x) = x'Ax, where A is symmetric independently of Q

[@proloycodes]

nice animations!

[@omridrori3286]

Can you make also on backprop ? But please do with matricess

[@wokeclub1844]

Excellent video illustration. But can someone tell me how at 8:23 he gets eigenvalues 1+x and 1-x?? I got them as 1-/+ sqrt(2*x) Pls help!

[@VisuallyExplained]

Thank you! It’s hard to point you in the right direction if you don’t explain how you found these eigenvalues, but Here is a sanity check you can do: the sum of the eigenvalues should be equal to the trace (i.e., the sum of the diagonal elements), and the product of the eigenvalues should be equal to the determinant.

[@jimlbeaver]

Wow, a light bulb for me turned on with this…thanks very much!

[@VisuallyExplained]

Great to hear!!!

[@yahavx]

What's the visual meaning of x^T*A*x?

[@NoNameAtAll2]

is there a name for "good solution to NP-hard problem" problems?

[@YahyaElakkazi]

Wa 7miiiiiim9

[@hamzaazeddagh1019]

Hadchi zine

[@YahyaElakkazi]

Maymknch

[@G0tBlackOps]

You speak to fast