So, the eigenvectors of a circular matrix are roots of unity -> Fourier matrix. I've used Fourier Transform and DCT in image processing and found it amazing but mysterious: where did it come from? Now I know. I also never realized convolution is essentially polynomial multiplication... and circular convolution simply applies a modulus to the powers! Mind expanding; thanks Prof Strang!
One of the most amazing thing about MIT classes is, no matter how difficult questions peofesssor ask in a class, there are students who knows the answers.
Sorry to be offtopic but does anyone know a tool to get back into an instagram account?? I stupidly forgot the login password. I would appreciate any help you can offer me
@Azariah Ivan Thanks so much for your reply. I found the site on google and I'm trying it out now. Looks like it's gonna take quite some time so I will get back to you later with my results.
Dr. Strang I see this matrix from a finite point of view. W is a generator 7 and P is 49 with a range 1 to p-1. 7^8 = 5764801, 5764801 / 48 = 120100.0208, 120100.0208 - 120100 = .0208 3333, .02080333 (48) = 1. I suppose this matrix is part of a discrete log problem. finding the exponent in Zp^*. this matix looks awfully close to the array for finding primitive roots. If so would numerical sequences that do not have a repeating pattern of numbers be Eigenvalues. Could you bring this cluster mess all together and show the short cuts.
So, the eigenvectors of a circular matrix are roots of unity -> Fourier matrix. I've used Fourier Transform and DCT in image processing and found it amazing but mysterious: where did it come from? Now I know. I also never realized convolution is essentially polynomial multiplication... and circular convolution simply applies a modulus to the powers! Mind expanding; thanks Prof Strang!
to be precise, the eigenvalues of the above circular matrix are the roots of unity, Fourier matrix = eigenvectors corresponding to those eigenvalues
Amazing! It's a real joy to follow Prof. Strang's lectures.
One of the most amazing thing about MIT classes is, no matter how difficult questions peofesssor ask in a class, there are students who knows the answers.
Gilbert Strang, you have done so much to help me learn linear algebra for my research. Thank you!!
Sorry to be offtopic but does anyone know a tool to get back into an instagram account??
I stupidly forgot the login password. I would appreciate any help you can offer me
@Drew Sam instablaster ;)
@Azariah Ivan Thanks so much for your reply. I found the site on google and I'm trying it out now.
Looks like it's gonna take quite some time so I will get back to you later with my results.
@Azariah Ivan It did the trick and I now got access to my account again. I am so happy:D
Thank you so much, you saved my ass !
@Drew Sam Glad I could help :D
Thank you so much for the beautiful ideas behind the circulant matrices. Geometric ideas help me to understand the concepts easily, thank you.
Prof Strang, thank you so much for all your lectures.
The legend! Thank you!
Professor Strang, thank you for explaining the Eigenvectors of Circulant Matrices ,Fourier Matrix and their impact on Machine Learning.
why do you just copy and paste the title on every single one of these videos?
@@amesoeursI don’t see there is a problem, he just wants to leave comment on each video to support the professor’s channel by boosting the algorithm
Interesting facts about normal matrices!
at 51:00 did he mean e^(8*pi/3) which would be 1 instead of e^(6*pi/3)?
No! is in facto because
e^(6 pi/3) =e^(4 pi/3)*e^(2 pi/3)=e^(4 pi/3+2 pi/3)
Dr. Strang I see this matrix from a finite point of view. W is a generator 7 and P is 49 with a range 1 to p-1.
7^8 = 5764801, 5764801 / 48 = 120100.0208, 120100.0208 - 120100 = .0208 3333, .02080333 (48) = 1. I suppose this matrix is part of a discrete log problem. finding the exponent in Zp^*. this matix looks awfully close to the array for finding primitive roots. If so would numerical sequences that do not have a repeating pattern of numbers be Eigenvalues. Could you bring this cluster mess all together and show the short cuts.