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:
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!

[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.

[piyukr]

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

[MayankGoel]

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.

[krishnasinghrajput]

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

[tiga]

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

[moopoo]

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

[abdulazizalhaidari]

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

[mspy]

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

[anjishnu]

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.

[ClearMath]

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!

[pauledam]

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

[praticandomatematica]

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!

[tylerperini]

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

[vasanthakumarg]

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

[michaelho]

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

[장발의둠피]

Please keep making videos. So helpful and intuitive!

[josecarlosferreira]

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

[무야호-wg]

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!!!!