Positive Definite Matrices

MIT RES.18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015
Instructor: Gilbert Strang

A positive definite matrix has positive eigenvalues, positive pivots, positive determinants, and positive energy.

License: Creative Commons BY-NC-SA

[mohamedakrout]

"Can I draw a picture ?", "Can I show you why it is the case ?". I simply love this man!

[robby.costales]

At 15:01 he mentions completing the square to show the energy expression is indeed positive for all values of x, but doesn't do it out in the video. It's actually quite simple to show all solutions are positive this way:
1. Our polynomial is: 4x^2 + 12xy + 10y^2. (I've replaced the x2 with y to make it neater).
2. We can also write this as: (4x^2 + 12xy + 9y^2) + y^2. (I've simply broken up the 10y^2 term).
3. We factor: (2x + 3y)^2 + y^2.
4. We know x and y are real numbers, so this sum of squares must be positive.
Hopefully this is helpful.

[MertTheDragonReborn]

I can't thank enough for the video. It is very useful and explained perfectly. Love from Turkey

[Sundaresan]

Explained beautifully, you are not just knowledgeable but also a great teacher, thanks a lot sir for explaining with such simplicity

[ToniSkit]

Gilbert Strang is the best- I really enjoy his classes! Awesome professor

[georgesadler]

Another great lecture by Dr. Gilbert Strang. Positive Definite Matrices are heart of linear algebra.

[justpaulo]

17:36 "going to Wall Street... becoming rich.... They don't give you all the money right away..." 😂

[Leowlion]

Thank you so much, sir. Real passion for mathematics this is.

[kejisong]

Why any of those 5 tests is sufficient to make the matrix S positive defined？We can still find a matrix A whose principle minors（the upper left determinant case you said）are positive but X^TAX less than or equal to 0 which means that A is not positive defined.

[liwang]

I am gonna learn linear algebra with him again!!!

[anas.k]

Hi, about the last exercise that you gave, ok, you gave a test that a, b, c have to verify but who ensure to us that there is not other a, b, c which could make the matrix definite positive. Thank you, Anas from Morocco.

[Tyokok]

great video! can some one point me to the link to the previous lecture? thanks a lot!

[AdityaPrasad]

11:57 So what's the borderline case for the 4rth test? all except ONE column of A is independent?

[austinputz]

I'm still fuzzy on the practical implications of PD matrices. It's like a circular argument, we make statements about all eigenvalues are positive, etc, but in practical breeding (we are animal breeders) I struggle to find implications for students. I googled this for some time a year or more ago and struggled to find anything online about implications. Just gives all the definitions but for OLS for instance or mixed models, are there many applications or implications we can point to for this being important to understand?

[hakansirin]

Can someone explain why ( A^T)A x = lambda x? It is in the very first board

[shrinivasiyengar]

I keep seeing everyone calling (x' S x) something related to energy. Can someone point me in a general direction as to what I should know/read about to know what that thing is that he is talking about. I understand that the topic has nothing to do in particular with this, but i would like to know more.

[donm]

thank you for the video, are we missing video 7.1?

[sethsetse]

Has anyone else noticed that he starts every single video with "Okay, uhh"

[markgeanacopoulos]

Great teacher, but it was quite disappointing to see advice at the end on how to ace an interview for a position as a wall street kleptocrat.