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

[@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!

[@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.

[@krishnasinghrajput4206]

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

[@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.

[@tiga7659]

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

[@tsunningwah3471]

😅😮😢🎉

[@mspy2989]

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

[@VisuallyExplained]

Awesome!!!! Good luck for your exam!

[@abdulazizalhaidari7665]

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

[@pauledam2174]

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

[@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. :)

[@yumika_pei]

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

[@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!

[@장발의둠피]

Please keep making videos. So helpful and intuitive!

[@VisuallyExplained]

Thank you! Will do!

[@bitflop]

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

[@VisuallyExplained]

Thank you Ario!!

[@michaelho5138]

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

[@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.

[@arcmchair_roboticist]

Wow this is an amazing video and a great channel!

[@VisuallyExplained]

Thank you so much!!

[@lithiumdeuteride]

Excellent video. Clear, concise, and illuminating.

[@VisuallyExplained]

Much appreciated!!!

[@srisaisubramanyamdavanam9912]

man you are a legend. you have got a new subscriber

[@fadi.almasalmah]

what a wonderful explanation! keep going!

[@amirreza08]

Just WOW! What a great explanation.Thank you!

[@MaiNguyen-yp1sj]

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

[@VisuallyExplained]

You're very welcome!

[@MohammadRamadan-o5c]

Brilliant explanation!

[@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.

[@jonasulmen5332]

huge fan!

[@VisuallyExplained]

Hooray! :)

[@priyankajain1691]

Nicely explained. Thank you

[@mariestrand9918]

Why did you place the z>=0 plane at such an angle in 8:00? I would have assumed it to be paralell to the xy-plane.

[@potatooflife8603]

this is amazing. thank you.

[@accountname1047]

This is a fantastic video

[@VisuallyExplained]

This is a fantastic comment!

[@CHUAN-CHI]

Absolutely wonderful.

[@jayantnema9610]

totally love your work dude!

[@gorbzzz]

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

[@무야호-w8g]

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!!

[@helloram300]

At 5:30 you say if one or more of the diagonal entries is zero then it is only positive semi-definite, but don't the other non-zero diagonal entries make the overall dot product of x and Ax greater than zero. Then it meets the criteria of positive definite right? What am i missing here? Thanks in advance.

[@tobiherberts3128]

If you have a matrix A that has a Zero as its i-th diagonal element, the multiplication v^T Av with v as the i-th canonical basis vector turns out to be zero ( it gives a exactly the ith element of the diagonal). So a matrix with a zero on its diagonal can at most be positive semidefinit. Analogously you can argue that a matrix cannot be psd when it has a negative number on its diagonal. Hope that helps

[@helloram300]

@@tobiherberts3128 thank you for responding to my question! It makes sense now. In the definition i saw 'for all x not equal to zero' so did not think of the canonical vector.

[@ZhanCaitao]

Nice video!

[@pp-xc8kb]

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

[@VisuallyExplained]

Thanks! Glad this was helpful!

[@danielx1912]

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

[@VisuallyExplained]

Thank you!

[@georgesms]

WOW! THAT WAS AWESOME!

[@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

[@stephaniedsouza3042]

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

[@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.

[@bilenkeziban6237]

Great video, thank you!

[@ocamlmail]

Tremendously cool explanation, thank you very much!

[@VisuallyExplained]

You're very welcome!

[@davidsal8308]

Awesome video!

[@fishinker]

So good, thank you

[@jkgan4952]

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

[@sebastianmezaurotadze2581]

awesome video

[@omridrori3286]

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

[@proloycodes]

nice animations!

[@yahavx]

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

[@jimlbeaver]

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

[@VisuallyExplained]

Great to hear!!!

[@NoNameAtAll2]

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

[@khushaalnandwani3372]

CSC420 gang

[@YahyaElakkazi]

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[@hamzaazeddagh1019]

Hadchi zine

[@YahyaElakkazi]

Maymknch

[@G0tBlackOps]

You speak to fast

[@andres_camarillo]

Amazing video. Thanks!