@@nicolasmorabito7950 yeah same situation here. we were taught to compute it without any explanation of its use. Was just kinda dumped on us. I think the JCF should be learned in later Linear Algebra courses, not at the beginning
This used to seem so confusing. But you make it so simple, so clear. Congrats. You are very good explaining this kind of subjects. You show, indeed, the bright side of mathematics.
This is the best video on the Topic out there! Could you make a video, in the same fashion, about Orthogonal Matrix Decomposition? That would be amazing. Thanks a lot for your videos.
Great videos about Jordan form❤ I only learned a few about it in university since it is the last chapter of my textbook. Thanks you so much that I can review and learn that after many years!!
Really great! I am teaching my class this material, and I would just point them to your video, but, the module notes do not show the Jordan Chains in the same way. Alas! I will make my own video using your example, but give your 4 videos of JNF as metadata for the video. My video in any case will not be as fantastic as this!!
Unbelievably good video series. You made this topic so simple. Well done! I would also like to say that either of the 2 vectors in level 2 that are not in level 1 can work as choices for u2 for the calculation of u1. I tried using the one not used in the video, went through the computation and found that this alternate slightly different X matrix also satisfies the equation J=X^-1 A X.
if I were to add anything to this series, I would add a video showing how Jordan forms can be used to solve practical problems in the real world. I am in grad school finishing my Master's (and trying to move forward with the PhD) and I still have not had a practical reason to use this knowledge (besides tests). I'm sure I can find an example on my own, but at least for others this really should be part of a standard curriculum.
I am an undergraduate student and I have also struggled to find the usefullness of Jordan form. We've just learnt about systems of differential equations where this comes into play via matrix exponential. I think this is a nice example of application.
hi, thanks for your videos they are really helpfull! I just don't know how do you find (how to find) the span and the two new eigenvectors at 6:28. Thank you very much!
I am enjoying these videos very much and i had one question. namely, when calculating for eigenspace, you said that the first kernel is the span of those two vectors, but where did the second vector in the span bracket come from?
Just a comment. Love your videos, but could you use a larger cursor? Sometimes the explanation goes something like "you take this here and combine it with this here" but it's not clear where "here" is! I think you're relying on the cursor, but when it's just a dot it's hard to follow. Otherwise, great work.
Thanks! When you look at my newer videos, this problem shouldn't exist anymore. Can you check that? I would find it helpful to know your opinion there :)
When you're saying this like "this one here, etc", your pointer is too small to see and know which portion of the screen you're pointing too. There must be some way of making the pointer bigger I guess. Great explanations though! This didn't cause a problem here as the context is clear, but sometimes it may cause problem in complicated scenarios.
Hope this reply isn't too late! You can find the eigenvectors by seeing which 5x1 matrix multiplied by ker(A-lambda I) will give you a column of zeros. So the first vector (1,0,0,0,0)^T will result in all zeroes and so will (0,0,1,1,0)^T as this will cancel out the 1 and -1 entries in ker(A-lambda I).... not sure if I made my explanation clear
Just save me from UIUC MATH285, many thanks. The lecturer in this class just begin the jordan canonical form without even introducing linear algebra!!!
Thank you, these Videos are awesome! I have a question though: If we have a Mixture of your Examples 3 and 4, so two different Eigenvalues [ let's say (1-λ)(2-λ)^4 ] with γ(λ_2)=2 how would we determine the size of the Jordan-Boxes in the second Jordan-Block? (to form the Jordan Normal Form) Do we just use the bottom-right 4x4 part of A or the whole matrix when calculating the kernel of the second level? Because to me if you take the whole matrix you end up one dimension too high but it seems wrong to take just a part of the matrix. I hope that I could frame my question well - thank you in advance!
Thanks! And thank you very much for the support! For the Jordan normal form we consider different eigenvalues separately. So there is no real mixture :)
Oh, I really thought the magic vanishes when you see this procedure because it just fixes the flaws of a diagonalisation approach. However, you might be right: I should do a video about invariant subspaces and a proof of the whole Jordan normal form.
Edit: I get the length concept on video three. I had to watch the video several times because my slow mental processing speed often hinders my progress. Indeed, I missed your finger pointing motions during video three.
QUESTION: for the detection of the eigenvalue could we just multiply row 4 with (-1) and then exchange it with row 3 so that the matrix is upper diagonal, so the eigenvalues are all on the main diagonal?
How to choose vectors if geometric multiplicity is 1(i.e dim(ker(A-cI))=1) and algebraic multiplicity= dim(ker(A-cI)^2)=5. Now how to choose?like if we take 2 vectors from eigen space they will be L.D .not form basis
Also the k(A)= -(lambda)*( lambda-1)^4, after we calculate the determinant if I am not mistaken. Would the example work the same way or does the approach change?
Am I correct in assuming that the order in which the (generalized) e-vectors come in the transformation matrix must be in the same order as the Jordan blocks which are ordered by different e-values? To elaborate, in this example, there was one Jordan block for the e-value 1 of size 5, with Jordan boxes ordered 3x3 and 2x2. As such, the order of the (generalized) e-vectors in the transformation matrix was w1, w2, w3, corresponding to the 3x3 Jordan box, and u1, u2, corresponding to the 2x2 Jordan box. The question: if there was another e-value, say of 2, with a Jordan block size (arithmetic multiplicity) of 4, and its Jordan chains were v1, v2, v3, and x1, then would the order of the transformation matrix have to be: X=[w1,w2,w3,u1,u2,v1,v2,v3,x1], corresponsing to the Jordan normal form of: J=[J_3(λ_1), J_2(λ_1), J_3(λ_2), J_1(λ_2)] ? * *J_n(λ_i) denotes a Jordan box of size n for the e-value λ_i.
After watching all 4 videos on this I still don't know .... why do you want to do this? Yes, you've explained the factorisation, but what is the use of this factorisation?
Calculating powers of the matrix is very simple when you have the Jordan normal form. Powers of large matrices occur often in applications. For example, in differential equations.
@@brightsideofmaths Thanks for the quick response. Following up: (i) How do you quickly take large powers of non-diagonal matrices, even if they are close to diagonal? (ii) Even though you've prescribed an algorithm, I can't make the same link to geometry that I can with diagonalisation. I have the feeling that the name "normal form" should be telling me something, but I'm not sure what that is. Have you made any other videos which plug the gap between basic diagonalisation and this work?
Sorry, the video is in english, but the legend is configured for germany. It´s not good because there are problems to automatic translate for other languages.
What do I do if one of the Jordan chains is of length 1? E.g. level 1 has 2 dots and level 2 only has 1? I'm not sure which vector to use for the transformation matrix in this case. Thanks.
Use the lone generalized e-vector from level 2 (call it w_2), generate the e-vector of level 1 by multiplying w_1=(A-λ•I)•w_2. That's one Jordan chain corresponding to a Jordan box of size 2 within your Jordan block of size 3. Now pick an e-vector from level 1, which is not already in the span of the other (generalized) e-vectors, call it u_1. This corresponds to a Jordan box of size 1 within your Jordan block of size 3. Now when you order these (generalized) e-vectors in your transformation matrix as such: X=[..., w_1, w_2, u_1,...], you need to remember to order the Jordan blocks in your Jordan normal form as such: J=[...,Jordan block of size 2, Jordan block of size 1,...].
i think if the second case was true, we would have a jordan normal form with the 2x2 box first then the 3x3 box. both are solutions but he simply chose to put the 3x3 box first.
You mean they are complex but you want to stay in the real numbers? Then the thing will not work. This setup is for you working completely in the complex numbers.
Yeah, for an example. Do you have any videos that help me if I have eigenvalues in the complex but wanna stay in the real numbers and still wanna create a jnf?
![Jordan Normal Form - Part 1 - Overview [dark version]](http://i.ytimg.com/vi/WuVzj9jie84/mqdefault.jpg)
All this was weirdly left out of my linear algebra course. Thanks man, just became a fan. Keep doing what you're doing mate
yeah, feel the same way.
Maybe they didn't know it themselves
You’re lucky. It was thrown into mine with barely any explanation and we are expected to know it flawlessly with high abstraction.
@@nicolasmorabito7950 yeah same situation here. we were taught to compute it without any explanation of its use. Was just kinda dumped on us. I think the JCF should be learned in later Linear Algebra courses, not at the beginning
This used to seem so confusing. But you make it so simple, so clear. Congrats. You are very good explaining this kind of subjects. You show, indeed, the bright side of mathematics.
Finally someone who knows how to explain it, and finally I've learned how to do it. Thank you very much.
Great to hear! Thanks!
This is the best video on the Topic out there!
Could you make a video, in the same fashion, about Orthogonal Matrix Decomposition? That would be amazing.
Thanks a lot for your videos.
I finally understood it afer three whole days of struggle, i cannot thank you enough you were more helpful than i can explain
Great to hear! Thanks!
You just saved my assignment. Thanks
Happy to help! And thanks for your support!
currently studying for my final, your videos helped me grasp this. thanks a lot!
This was really really helpful! Thank you a lot man, i couldn't find any simple explanation of this topic on any book: you really saved the day!
Super fast, concise, and practical examples with great points being made throughout. Really appreciate the videos!
Thank you very much for your kind words and your support :)
Мужик, ты спас мне жизнь!
Man, you saved my life!
So high quality
Great videos about Jordan form❤ I only learned a few about it in university since it is the last chapter of my textbook. Thanks you so much that I can review and learn that after many years!!
You're so welcome! And thank you for your support for the channel :)
thankyou very much youre a star! you really helped a lot💖
You saved me! Thanks a lot from South Korea
Really great! I am teaching my class this material, and I would just point them to your video, but, the module notes do not show the Jordan Chains in the same way. Alas! I will make my own video using your example, but give your 4 videos of JNF as metadata for the video. My video in any case will not be as fantastic as this!!
Thanks for sharing! I hope that it really helped your students :)
Just fantastic! Beautifully done. Thank you.
You are welcome :)
Unbelievably good video series. You made this topic so simple. Well done!
I would also like to say that either of the 2 vectors in level 2 that are not in level 1 can work as choices for u2 for the calculation of u1. I tried using the one not used in the video, went through the computation and found that this alternate slightly different X matrix also satisfies the equation J=X^-1 A X.
Wow that was a big help!! thank you very much!
if I were to add anything to this series, I would add a video showing how Jordan forms can be used to solve practical problems in the real world. I am in grad school finishing my Master's (and trying to move forward with the PhD) and I still have not had a practical reason to use this knowledge (besides tests). I'm sure I can find an example on my own, but at least for others this really should be part of a standard curriculum.
I am an undergraduate student and I have also struggled to find the usefullness of Jordan form. We've just learnt about systems of differential equations where this comes into play via matrix exponential. I think this is a nice example of application.
Sweet video
Thanks a lot man saved me
what a life saver! thank you
My pleasure :)
You can also download the pdf version here: tbsom.de/s/ov
Thanks professor
Very informative and easy to understand, thank you!
You are very welcome :)
thank you a lot! The best explanation Ive seen so far
Thank you very much!
Gracias.
You are welcome :)
Your videos are so good!
thanks for the great video
thank you
hi, thanks for your videos they are really helpfull! I just don't know how do you find (how to find) the span and the two new eigenvectors at 6:28. Thank you very much!
I am enjoying these videos very much and i had one question. namely, when calculating for eigenspace, you said that the first kernel is the span of those two vectors, but where did the second vector in the span bracket come from?
I L❤VE IT!
Just a comment. Love your videos, but could you use a larger cursor? Sometimes the explanation goes something like "you take this here and combine it with this here" but it's not clear where "here" is! I think you're relying on the cursor, but when it's just a dot it's hard to follow. Otherwise, great work.
Thanks! When you look at my newer videos, this problem shouldn't exist anymore. Can you check that? I would find it helpful to know your opinion there :)
When you're saying this like "this one here, etc", your pointer is too small to see and know which portion of the screen you're pointing too. There must be some way of making the pointer bigger I guess. Great explanations though!
This didn't cause a problem here as the context is clear, but sometimes it may cause problem in complicated scenarios.
In the newer videos I fixed the pointer problem :)
@@brightsideofmaths oh okay 😅
@@ar3568row Sadly, I didn't realise earlier how to increase the pointer :D
really thanks sooo much, in fact, you really helped a lot. Could you plz, make videos for the Decomposition matrix???? plzzzzzz
You are welcome! What do you mean by decomposition matrix?
@@brightsideofmaths matrix factorization and Decomposition, this is my mean
from LU to LDV
I don’t understand how we found the eigenvectors
Hope this reply isn't too late! You can find the eigenvectors by seeing which 5x1 matrix multiplied by ker(A-lambda I) will give you a column of zeros. So the first vector (1,0,0,0,0)^T will result in all zeroes and so will (0,0,1,1,0)^T as this will cancel out the 1 and -1 entries in ker(A-lambda I).... not sure if I made my explanation clear
Just save me from UIUC MATH285, many thanks. The lecturer in this class just begin the jordan canonical form without even introducing linear algebra!!!
You are welcome! :)
king
Thank you, these Videos are awesome!
I have a question though: If we have a Mixture of your Examples 3 and 4, so two different Eigenvalues [ let's say (1-λ)(2-λ)^4 ] with γ(λ_2)=2 how would we determine the size of the Jordan-Boxes in the second Jordan-Block? (to form the Jordan Normal Form)
Do we just use the bottom-right 4x4 part of A or the whole matrix when calculating the kernel of the second level? Because to me if you take the whole matrix you end up one dimension too high but it seems wrong to take just a part of the matrix.
I hope that I could frame my question well - thank you in advance!
Thanks! And thank you very much for the support! For the Jordan normal form we consider different eigenvalues separately. So there is no real mixture :)
Thank you for the great explanation! Do you have a video on why this works? Not a proof, but an intuition behind this magic.
Oh, I really thought the magic vanishes when you see this procedure because it just fixes the flaws of a diagonalisation approach. However, you might be right: I should do a video about invariant subspaces and a proof of the whole Jordan normal form.
Could you make a video explaining more about this? I don't really get how to know the order of the vectors in the transformation matrix
But how did you derive the eigenvectors in Span ?
Calculating eigenvectors. I have some videos about that on my channel.
Edit: I get the length concept on video three. I had to watch the video several times because my slow mental processing speed often hinders my progress. Indeed, I missed your finger pointing motions during video three.
You mean the different levels for building the Jordan chains?
I think that I get the idea. I enjoy watching videos because I always stumble on the first viewing. Good work!
I don't understand how you decide where the red 1's go in the J matrix
QUESTION: for the detection of the eigenvalue could we just multiply row 4 with (-1) and then exchange it with row 3 so that the matrix is upper diagonal, so the eigenvalues are all on the main diagonal?
How to choose vectors if geometric multiplicity is 1(i.e dim(ker(A-cI))=1) and algebraic multiplicity= dim(ker(A-cI)^2)=5. Now how to choose?like if we take 2 vectors from eigen space they will be L.D .not form basis
Is there any reason why we would be interested in X since we don't need X to determine J ?
For calculations, we still need X to transform between A and J.
At 4:35. How exactly did you calculate that Ker( A- lamda*I) = span 《 ( 1,0,0,0),
( 0,0,1 ,1 ,0)》?
Also the
k(A)= -(lambda)*( lambda-1)^4, after we calculate the determinant if I am not mistaken.
Would the example work the same way or does the approach change?
I missed the info about the book ,if you have nentioned earlier.
Please ,let me know the book name.
The book is on my website: tbsom.de/com
Can someone explain in more detail why we get the ones above the diagonal of J? :)
спасиба!
You are welcome!
Hey could you please explain how the span was obtained at minute 4:09? I don't understand how that was obtained
Did you watch my Linear Algebra series? You learn everything there: tbsom.de/s/la
Can you explain how you got the vectors in the span
Yes, we solved the system of linear equations given by the matrices.
could u explain for case with linear time variant A matrix
I could have started with u1 as well, right? as in X = (u1, u2, w1, w2, w3)
Other than that this video was amazing, thanks a lot!
Yeah, of course. But then the Jordan normal form is also different.
@@brightsideofmaths Thanks for the quick reply! Left you a subscription prior to my earlier comment.
Am I correct in assuming that the order in which the (generalized) e-vectors come in the transformation matrix must be in the same order as the Jordan blocks which are ordered by different e-values?
To elaborate, in this example, there was one Jordan block for the e-value 1 of size 5, with Jordan boxes ordered 3x3 and 2x2. As such, the order of the (generalized) e-vectors in the transformation matrix was w1, w2, w3, corresponding to the 3x3 Jordan box, and u1, u2, corresponding to the 2x2 Jordan box.
The question: if there was another e-value, say of 2, with a Jordan block size (arithmetic multiplicity) of 4, and its Jordan chains were v1, v2, v3, and x1, then would the order of the transformation matrix have to be: X=[w1,w2,w3,u1,u2,v1,v2,v3,x1], corresponsing to the Jordan normal form of: J=[J_3(λ_1), J_2(λ_1), J_3(λ_2), J_1(λ_2)] ? *
*J_n(λ_i) denotes a Jordan box of size n for the e-value λ_i.
Yes, the order is important. You can look at my book/lecture notes for more details.
@@brightsideofmaths Thanks
How do you get the span in the ker(A-1I) : time stamp 4.17
I calculated it like here: thebrightsideofmathematics.com/courses/calculating_kernel/overview/
And here: thebrightsideofmathematics.com/courses/calculate_dimension/overview/
After watching all 4 videos on this I still don't know .... why do you want to do this?
Yes, you've explained the factorisation, but what is the use of this factorisation?
Calculating powers of the matrix is very simple when you have the Jordan normal form. Powers of large matrices occur often in applications. For example, in differential equations.
@@brightsideofmaths Thanks for the quick response. Following up:
(i) How do you quickly take large powers of non-diagonal matrices, even if they are close to diagonal?
(ii) Even though you've prescribed an algorithm, I can't make the same link to geometry that I can with diagonalisation. I have the feeling that the name "normal form" should be telling me something, but I'm not sure what that is. Have you made any other videos which plug the gap between basic diagonalisation and this work?
how did you calculate the span of the kernel?
2:00 Can someone help me find the eigenvalues? I don't understand how to do it. I interchanged row 3 and 4 but there''ll be a -1. That's not right 😭
ok i solved it myself. Just follow the good ol procedure for finding determinant.
what about the eigenvalues 0 and 2 for this matrix at the beginning?
Why do you think that 0 and 2 are eigenvalues?
For two different eigen values how to find the transformation matrix?
You do the steps in the video, two times.
@@brightsideofmaths different transformation matrix for different eigen values?
@@rajashreemukherjee3203 No, just a combined one.
@@brightsideofmaths did not understand
@@rajashreemukherjee3203 Take a look at my Linear Algebra book :) (See description)
Sorry, the video is in english, but the legend is configured for germany. It´s not good because there are problems to automatic translate for other languages.
Thanks! I will fix it!
can someone please explain how he got the characteristic polynomial so fast at the start?
I skipped the calculation in this video because it is not so hard. You can do it!
what if there are 2 eigenvalues?
Then you need to this for each eigenvalue :)
@@brightsideofmaths but will your jordan basis still be a set of 4 vectors? which would you choose?
What do I do if one of the Jordan chains is of length 1? E.g. level 1 has 2 dots and level 2 only has 1? I'm not sure which vector to use for the transformation matrix in this case. Thanks.
Use the lone generalized e-vector from level 2 (call it w_2), generate the e-vector of level 1 by multiplying w_1=(A-λ•I)•w_2. That's one Jordan chain corresponding to a Jordan box of size 2 within your Jordan block of size 3.
Now pick an e-vector from level 1, which is not already in the span of the other (generalized) e-vectors, call it u_1. This corresponds to a Jordan box of size 1 within your Jordan block of size 3. Now when you order these (generalized) e-vectors in your transformation matrix as such: X=[..., w_1, w_2, u_1,...], you need to remember to order the Jordan blocks in your Jordan normal form as such: J=[...,Jordan block of size 2, Jordan block of size 1,...].
Hi !
At 6:27 , how did you calculate the eigen vectors ? its a little confusing
Just calculate the kernel like always. I have a video about this :)
Why X=[ w1 w2 w3 u1 u2 ] instead of X=[ u1 u2 w1 w2 w3 ] ?
i think if the second case was true, we would have a jordan normal form with the 2x2 box first then the 3x3 box. both are solutions but he simply chose to put the 3x3 box first.
What should we do if there is 1 eigenvector on level 1 and 2 newly added generalised vectors on level 2. (how does the chain work in that case)
The question should be: Is that even possible?
@@brightsideofmaths I did get it, but it was probably a miscalculation (:
What do we do if the eigenvalues are not within the body?
You mean they are complex but you want to stay in the real numbers? Then the thing will not work. This setup is for you working completely in the complex numbers.
Yeah, for an example.
Do you have any videos that help me if I have eigenvalues in the complex but wanna stay in the real numbers and still wanna create a jnf?
@@QuamQuas It's just not possible in this sense anymore.
@@brightsideofmaths well, my linear algebra 2 homework wants me to calculate the jnf and the eigenvalues are not within Q although the matrix is.
@@QuamQuas I am sorry to hear about this. What is your characteristic polynomial?
could as you, why are you stopped at order 3 , how can could the order for the calculation, could you tell me plzz
as at the order 3 the dimension of ker is 5 which is same to Algebraic Multiplicity .
can someone explain how we get the eigen vectors? in simple words please
:)