Quadratic Form Minimization: A Calculus-Based Derivation

[@MathTheBeautiful]

Go to LEM.MA/LA for videos, exercises, and to ask us questions directly.

[@snnwstt]

1:18 Just as an observation, while it is usual to see the quadratic form as presented here, I find the following a little bit more ... elegant: 0.5 * [W] {x y z 1} With a line vector, { } a column vector and [ ] a matrix. Here W = 4 1 2 -2 1 8 5 -3 2 5 4 -4 -2 -3 -4 0 symmetric if A is symmetric. Note that the minus sign for the last column and the last line is due to the original subtraction. The 0 stands when the constant term is ... zero.

[@vothiquynhyen09]

I have to say that I love your voice, and the passion you have for the subject.

[@joshuaronisjr]

He talks a little like Feynman

[@ekandrot]

For your gradient descent, do you need the -b in there, eg x -> x - a(Ax-b) ? It seemed without that -b and a positive definite matrix A, zero is the only solution. But with -b then -1,-2,4 is the solution.

[@MathTheBeautiful]

Yes, you are correct!

[@ijustneedaname47]

This video really helped tie these concepts together for me. I really appreciate your posting it.

[@omedomedomedomedomed]

To understand the least square derivation, I check this. Super helpful !!!

[@jesalshah81]

Omg, I have never learnt math like this before! So great!

[@MathTheBeautiful]

I'm glad you feel this way!

[@bryan-9742]

this is so cool. Love this channel. I'm learning so much I should have learned years ago.

[@sora290762594]

great way of explaining quadratic optimization

[@joaquingiorgi5133]

Made this concept easy to understand, thank you!

[@Userjdanon]

Great video. This was explained very intuitive.

[@TuNguyen-ox5lt]

Gradient descent is a technique used in machine learning nowadays to optimize a loss function . This video is great

[@gerardogutierrez4911]

Why does he talk like hes trying to get me to recapture the means of production from the bourgeoisie?

[@MathTheBeautiful]

Because he is lenin in that direction

[@jjgroup.investments]

Thanks for this awesome video

[@devrimturker]

Is there a relation between positive definite matrix and convex set

[@MathTheBeautiful]

Yes, excellent intuition. The level set for a positive-definite quadratic form is a convex shape.

[@DiegoAToala]

Thank you, so clear!

[@ibrahimalotaibi2399]

Monster of Math.

[@s25412]

7:15 what if your matrix is positive semi-definite? Wouldn't there be a minimum?

[@bobstephens97]

Awsome. Thank you.

[@MathTheBeautiful]

Thank you!

[@ashwinkraghu1646]

Excellent teacher! and Life saver

[@user-xt9js1jt6m]

Nice explanation sir You look like Jason Statham ❤️❤️❤️ I felt like action star is giving lecture on matrix❤️❤️🙏

[@MathTheBeautiful]

I get that a lot when I wear a tight t-shirt.

[@kreechapuphaiboon4886]

Great lecture he explains so well.

[@la_kukka]

Great explanation!! Thank You

[@MathTheBeautiful]

Glad it was helpful!

[@AliVeli-gr4fb]

thank you, it was a beautiful course

[@MathTheBeautiful]

Thank you, Ali, I'm glad you're enjoying our videos. But why "was"?

[@AliVeli-gr4fb]

MathTheBeautiful it is normal to say it in past tense in my language, so I thought in it, but wrote in English. so no real reason

[@MathTheBeautiful]

:) I just wanted to convey that the course is ongoing!

[@TheTacticalDood]

@@MathTheBeautiful Is it still ongoing? This channel is amazing, it would be sad to see it stop!

[@みの-c5c]

This really helps a lot in understanding matrix derivative, and it's so clear. Thanks!!!

[@somekindofbluestuff]

thank you!

[@serkangoktas5502]

I always knew that something was off with this derivation. I am relieved that this wasn't because of my lack of talent in math.

[@MathTheBeautiful]

It's **never** you. It's always the textbook.

[@johnfykhikc]

where i can found the statement ? i did an unsuccessful search

[@kaursingh637]

SIR - U R VERY CLEAR =PLEASE GIVE SHORT LECTUR

[@joshuaronisjr]

This is just a comment for me to look at in the future, but at some point, he says that A will mostly be filled with zeroes before we start Gaussian elimination. A will be the covariance matrix, (X^T X) (look at the next video, the least squares solution video). That it's mostly filled with zeroes indicates that most of the random variables (each column of X is a different random variable of the dataset) are independent of one another (or at least, if they ARE independent then their covariance will be 0). However, Gaussian elimination involves linearly combining rows. The matrices in between may NOT be sparse! As for computer storage...I don't know much about it, but maybe computers store zeroes in a different way, so that sparse matrices are easier to store? Actually, I guess this comment is more than for just me...why can computers store sparse matrices well?

[@telraj]

Why skip the matrix calculus? It's not rocket science

[@roaaabualgasim4882]

I wont examples or meteial to illusstrate the idea of method of maximization and minimaization of function with constraint(lagrage multiplier ) and with no constraint(by quadratic form and hessian matrex) 😭

[@marshall7253]

I love this guy

[@darrenpeck156]

Absolute value has a minimum.

[@gustavoexel5569]

At 13:15 my chin literally felt

[@ElizaberthUndEugen]

*dropped