Linear Algebra 16h6: Generalized Eigenvectors

[frankreashore]

Fantastic short lecture. Your blackboard is SO “clean”. Thanks. Reading Axler.

[RubberDuckyToy]

From 2:00 to 4:28, why did you dub over yourself?

[EngBandar]

People who got confused how he did compute [0 0 -1] as a generalized eigenvector.
First: Use Gaussian-Jordan for the augmented matrix
3 -2 -1 | 1
3 -2 -1 | 1
2 -1 -1 | 1
We get
1 0 -1 | 1
0 1 -1 | 1
0 0 0 | 0
Put them in equations
x -z = 1
y -z = 1
0 0 0= 0
Or
x = 1 + z
y = 1 + z
Notice z is a free variable. We could choose any value for it, so for simplicity, let z = -1 ---> x = 0, y = 0 hence, the generalized eigenvector is v=[0 0 -1] which it will be used for the next computation.

[EdwardNusinovich]

Thanks so much man, I was confused about how to find a specific generalized eigenvector and you really helped. Great videos.

[wwfycxx]

Thank you professor I was really confused in this part, this video helped so much

[erlanggaz]

Very old video yet still as phenomenal. Very easy to understand explanation!

[sudiptapatowary]

In this case, as there was only one eigenvector, we used that one to derive the generalized eigenvectors. What happens for a 3x3 matrix which has two eigenvalues, one with a multiplicity of 2 and the other with 1. Which of the two eigenvectors would we use to derive the other generalized eigenvector?

[chrisdesrochers]

Hello Sir. I haven't been watching your entire series, so perhaps this is why I don't recognize what you're referring to. How is the first generalized vector [0 0 -1] ^T found in the column space of the Matrix A? At least right now I don't see it. See 5min46second mark. -- Ok I see it.

Thanks for effort you put in. :)

[fanalysis]

Can someone explain to me what "matrix pencils" have to do with the generalized eigenvalue problem?

[johnlin]

I was wondering, what is the physical meaning of the generalized eigenvector?

[streetwolfe]

dude, thank you SOOO much! I'm currently taking Adv Lin Alg II for the summer (fast paced) and I had no idea how they found that other e.vector (gen. e.vector) Liking this and sharing it to my peeps! haha :D

[edwinwidjaja]

To find the generalised vector of rank 3 in this example, would it be possible to simply derive it from the cross product of the eigenvector and the generalised eigenvector of rank 2?

[bogusmcbogus]

Why couldn't [1, 1, 0] work as the first generalized evec? Why specifically did you select [0, 0, -1]? Thanks in advance.

[mystic]

i dont understand why s the eigenvector in the column space ? and what is the proof that this algorithm will always ?

[ElektrikAkar]

What if you want to go to [1;1;1] for the second generalized eigenvector? I mean, you target [0;0;-1], but let us target [1;1;1] again and we may find [1;1;0] which is not same as [0;0;-1]. Can we take this vector as the second generalized eigenvector?

[xueqiang-michaelpan]

Could you please help explain why the null space is contained in the column space? Thanks!

[vineetmukim]

Sir,
The eigenvectors of the defective matrix do not form a basis. So if (as shown in this lecture) we have one eigenvector, we can always arbitrarily choose other two vectors which will be LI and form a right-handed system (using dot and cross products). What is the need of going through this procedure if all we wanted was a basis with originally found eigenvector as one of the vectors of basis? I am not getting the use/application of these specially found generalized eigenvectors. Please clarify.

[artun]

Sir, is it possible for us to determine the Jordan Form of the 3x3 given here, by using the P matrix( where P = [v1, v2, v3] & v1=eigvec AND v2 and v3 are the generalized eigen vector which are found in this video-)
J=inv(P)*A*P---->If the answer is yes, I did it and get the the matrix as following,
[0 1 0
0 0 1
0 0 is, Why is that?
Aren't we suppose to get the matrix as following,
thanks in advance.
[3 1 0
0 3 1
0 0 3]

[madhuriarya]

how do you know that the null space of the matrix (A-3I) is one-dimensional, as you stated directly without giving any arguments that lead to this conclusion.

[ruddha]

Are you Paul Scheer’s smarter brother?