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.
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!
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.
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
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!!!!
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!
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.
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
@@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.
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.
Thank you for making this, finally I get a concrete, visual intuition of what positive definiteness means!
You are very welcome. Thanks for the nice comment!
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.
Thank you so much for the very nice comment, stay tuned for more!
How and why is this channel so underrated. The quality of work is one of the finest.
Finally I got a geometric intuition of what positive definiteness means. Very well explained! Thank you so much.
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.
Amazing video! After years of studying at uni I finally understood the real intuition behind Positive definite matrices. Thank you so much! :D
😅😮😢🎉
What a great effect your videos have on the course of understanding of many people,
Lol you literally uploaded this video the day before my convex optimization exam. Thanks!
Awesome!!!! Good luck for your exam!
This is a BEAUTIFUL video. I am floored by your animations and how you communicated the relevance of PSD matrices.
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.
Thank you very much. :)
Boom! What a great way to start out -with a single element matrix were the element is a positive number. Fantastic!
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!
Thank you so much 😀 I checked your channel and you are doing such fine work yourself!
Such a great explanation! I have never read a more clear definition of positive definite matrix.
Please keep making videos. So helpful and intuitive!
Thank you! Will do!
This is an amazing series, and I look forward to sharing this with my students!
Please do!
Such beautiful and clear explanations. You are a legend in the making!
Thank you Ario!!
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.
This is a really great video-love the vibe from the music and timing at the end 😂
man you are a legend. you have got a new subscriber
Wow this is an amazing video and a great channel!
Thank you so much!!
wow, best explanation on positive definite that I can find. Thank you so much
You're very welcome!
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
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!!!!
Great to hear!!
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!
Explanation and animations are really great. keep up the good work.
huge fan!
Hooray! :)
Excellent video. Clear, concise, and illuminating.
Much appreciated!!!
This is amazing. I would love to see some "visual" explanation of SDP relaxations a la Lassere.
Just WOW! What a great explanation.Thank you!
what a wonderful explanation! keep going!
This is a fantastic video
This is a fantastic comment!
I cannot say anything more than amazing video! Thank you for your huge contribution!
Thanks! Glad this was helpful!
Amazing amazing videos, cannot say how much I appreciate this!
Thank you!
Absolutely wonderful.
Brilliant explanation!
this is amazing. thank you.
totally love your work dude!
Nicely explained. Thank you
I am sleepy and tired but this was still an awesome watch
Nice video!
WOW! THAT WAS AWESOME!
Tremendously cool explanation, thank you very much!
You're very welcome!
Amazing video. Thanks!
Awesome video!
So good, thank you
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.
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.
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
@@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.
awesome video
Great Video! Thank you so much :))))))))))))))))))
Wow, a light bulb for me turned on with this…thanks very much!
Great to hear!!!
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!
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.
Amazing
nice animations!
Can you make also on backprop ? But please do with matricess
What's the visual meaning of x^T*A*x?
is there a name for "good solution to NP-hard problem" problems?
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You speak to fast
Great video, thank you!