2-7 Givens rotations

Link to slides: www.dropbox.co...

[@turuus5215]

I subscribed! Martjin, you are an excellent teacher. Now, I'm gonna check your other videos. You teach some next-level topics.

[@ashar4121]

Watching from Zürich, this is so great! You should have more subscribers

[@lauragaleraalfaro8086]

This video really helped me to understand givens rotations. Thank you!

[@karol2100]

Great video, thank you for everything. Watching from Poland (saying this just to empemphasize that your video is helpfull around the world)

[@martijnanthonissen]

That's great to hear. Thanks for letting me know Karol!

[@Schooling2023]

this is so great! Thank's

[@TheLeontheking]

Small correction: at 11:00 the latter vector should be (sqrt(5), 0, 3) It's obvious from the equations listed afterwards though :)

[@martijnanthonissen]

Yes! You are right. Thanks for commenting

[@andrapi6024]

Great explanation! Thank you!

[@momentouscrazynoob1709]

Thank you so much, I finally understand this concept!!

[@martijnanthonissen]

Great to hear that. Thanks for your message!

[@mxyptlkk]

Thank you professor

[@jahanvi9429]

thank you very much , it was helpful

[@holyshit922]

Here it is easy to derive multiplications by Givens rotations and write it in the code When I mulitiplied A by G from left to get R i noticed that multiplication changes only two rows (labeled p and q in this video) When I mulitiplied I by G from right to get Q i noticed that multiplication changes only two columns (labeled p and q in this video) I also wrote this system of equations from which I got values for c and s Yes there is 1/2n(n-1) multiplications but single multiplication can be done just in one loop in O(n) time so both Givens rotations and Householder reflections are O(n^3) they differ only by constant If you try to compute roots of polynomial equations by eigenvalues method (it is numerical procedure and Givens rotations can help to find QR decomposition) (I know that it is space costly and there is a problem with repeated roots but some programs like Octave or packages like numpy use this approach) two of the companion matices are in upper Hessenberg form and two of them are in lower Hessenberg form In this case it seems that there is no difference between Givens rotations QR decomposition and Householder reflection QR decomposition

[@martijnanthonissen]

Thanks for your comment! You often see a preference for Givens rotations when a matrix is “almost” upper triangular: if you have a matrix in upper Hessenberg form (upper triangular plus one lower diagonal) then you can use n-1 Givens rotations to just make the lower diagonal zero

[@franciscoleandrosilvadelav174]

Thank you!

[@heidioid]

Bookmark guided example 9:33

[@fareoadewunmi8835]

Great lecture! Thank you

[@martijnanthonissen]

Glad you like it! Thanks!

[@takudzwamashamba7453]

woah thanks !!

[@oafccouchpotato]

very well explained

[@m.pranavphanindrasai2473]

It was very clear Sir Thank you . Which book did you take for reference ??

[@martijnanthonissen]

Thanks! I quite like the book "Scientific Computing" by Gander, Gander and Kwok

[@recognize1992]

thank you for the lecture :) I have one simple question. why givens rotation matrix is a form of [c s;-s c] instead of [c -s;s c] despite the latter is a rotation matrix?

[@martijnanthonissen]

You can do both: one is a rotation in clockwise direction, the other is a rotation counterclockwise. Or, formulated differently, one is a rotation over an angle phi, the other is a rotation over -phi

[@prvizpirizaditweb2324]

the only thing i did not understand was, why do we make g array that will not touch first , second or third entry, how this rule is applied?

[@martijnanthonissen]

Thanks for your question! Which slide or which time point in the video is it related to? I am not sure what it is that you are asking, so please let me know

[@user-dl9no7jb4b]

May I ask how the matrices of G1, G2, and G3 are formed? I don't quite understand the position change of c s -s c .

[@martijnanthonissen]

Thanks for your message! Say you would like to create a zero entry in the first column in row k. Then you need a rotation in the (1,k)-plane. The locations of the c, s, -s and c entries follow from the position (1,k). The main idea is that you make the (1,k) entry zero. However, you need to preserve the length of the first column, so you also change the (1,1) entry. Hope this helps!

[@user-dl9no7jb4b]

@@martijnanthonissen Thank you!! I got it!!

[@intansherlin4994]

Where could I find the slides on Givens rotations, Prof. Anthonissen?

[@martijnanthonissen]

I added a link to the slides in the description. Thanks for asking!

[@intansherlin4994]

Your welcome, Professor! Thank you for making linear algebra easier for me to comprehend 👍

[@ajayiakinola9672]

is there a mistake at G1.A row1-column 1? it should be 5/sqrt(5)?

[@martijnanthonissen]

That’s the same, right? In the video I write sqrt(5), which is 5/sqrt(5)