Symmetric Matrices, Real Eigenvalues, Orthogonal Eigenvectors

[@mirelladepietra9462]

Simply the best. How much passion in this man?how much charisma, knowledge power and skills???this is awesome,mesmerizing.

[@georgesadler7830]

This is linear algebra at its finest. Dr. Strang you are a linear algebra legend.

[@kamilazdybal]

Superb. Like all lectures of Professor Strang.

[@dalisabe62]

the minute you think you got bored watching him, he will hit you with a dose of refreshing challenge. He never swamps you with theory to the point of losing touch with reality, but he uses simple examples to extrapolate theory very quickly. Amazing lecturer who mastered his material yet never got tired of presenting it to different crowds.

[@mohammedal-haddad2652]

Professor Strang, every lecture of yours is like a symphony to me. Thank you very much.

[@MauAlexMX5]

Pure gold..

[@saitaro]

Quality of explanation: Prof. Gilbert Strang

[@Zuwwar]

very well done

[@pulkitatry157]

Incridible 😮.... that's a time i truly enjoyed, every second was full of knowledge.Such a marvelous teacher ❤

[@brianmirchin6990]

This man is a legend

[@gordito199916]

cosa bella , cosa hermosa , cosa bien hecha

[@alejandroencinas2905]

What is this guy talking about?, idk but this video was amazing!!

[@ucthuanphung4530]

Thank you Professor.

[@oliviakumar5903]

@2:22 is my favorite part!

[@wellingtonmusyoka7486]

Very elaborate and simple in explanation.

[@siddhartharaja9413]

Just wants to thanks,only because this content is free!!

[@Diana-yl1jo]

As he said at 4:12 the trace is 6 and determinant is 8, so the eigenvalues 2and4 is correct. Can anyone plz tell me what and why they have such relationship?

[@gainauntu]

det of that marix = 8 which should be equal to the product of eigenvalues which is 4*2=8.....since one the eigenvalue is 2...another on should be trace(MATRIX)-2=6-2=4

[@valeriapedrosa7964]

Because the Characteristic Equation of a Matrix A (2X2 ) is : P(x)= x^2 - trace(A)x + det(A)

[@BabaBoee5198]

Watch the video of 3Blue1Brown titled: A quick way to compute eigenvectors and eigenvalues. It’s definitely one of the best explanations you could find on the entire planet

[@peterbaker345]

the Passion!!!

[@abhishekmaurya6562]

Oh, I landed at the right video.

[@aziz-dailycommentsandmore9086]

ah I see, no men of culture here, excuse me

[@mastrammeena328]

Prooooooof please

[@jatinarora9463]

wonderful lecture by prof gilbert strang. Like the way he taught made it easy

[@edghar7995]

👑

[@allyourcode]

Isn't B an antisymmetric matrix? Doesn't that mean that the eigenvalues are supposed to be imaginary? But according to Professor Strang's calculation, the eigenvalues are 3 + i and 3 - i, which are not imaginary...

[@iteoluwaoladejo4240]

B is not antisymmetric.

[@mlabodia]

Why not?

[@sourishsarkar5281]

What if I have a symmetric matrix with repeated eigenvalues? Will the eigenvectors will always be orthogonal?

[@nickirpdark]

No, orthogonality only happens if the eigenvalues are different. For the case of repeated eigenvalues your space is degenerate.

[@sourishsarkar5281]

Nicolas .Pacheco Please refer to the book by Gilbert Strang which I came across recently, where he has proved by mathematical induction that the eigenvectors of a symmetric matrix will be orthogonal even if the eigenvalues are repeated..

[@nickirpdark]

which one, Introduction to Linear Algebra or Linear Algebra and Its Applications? and if it's not too much trouble can you remember the chapter.

[@sourishsarkar5281]

Introduction to linear algebra. I have the fourth edition. There, it is explained in chapter 6: Eigenvalues and Eigenvectors in section 6.4: Symmetric matrices.

[@nickirpdark]

The ideia here is that for symmetric matrices you can choose a suitable orthonormal basis that generates the eigenvector space., such that you can diagonalize the matrix The thing with repeated eigenvalues in symmetric matrices is that you have an eigenvector space of dimension n, where n is the multiplicity of of the eigenvalue, so it's possible for you to choose an pair eigenvectors that are not necesserally orthogonal.

[@zaeemlearninglab3154]

nice working

[@davidk7212]

It breaks my heart to say it, but this man needs to be put to sleep - he is *clearly* struggling and suffering. He absolutely is a national treasure and should forever be remembered as such, but it is time to give this man the dignified - and much overdue - outro that he so rightfully deserves and desperately needs.

[@owzok7087]

?????

[@bismeetsingh352]

What is an orthogonal matrix?

[@ajarivas72]

A is orthogonal matrix if Transpose(A) = A and determinant(A) = 1

[@AchtungBaby77]

@@ajarivas72 An easier way to define an orthogonal matrix is that its transpose is its inverse.

[@ajarivas72]

@@AchtungBaby77 You are correct. I made a mistake in my response. Transpose(A) = inv(A)