31. Eigenvectors of Circulant Matrices: Fourier Matrix

[@H2Obsession]

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!

[@exoticcoder5365]

to be precise, the eigenvalues of the above circular matrix are the roots of unity, Fourier matrix = eigenvectors corresponding to those eigenvalues

[@RAJIBLOCHANDAS]

Amazing! It's a real joy to follow Prof. Strang's lectures.

[@ethanjensen1564]

Gilbert Strang, you have done so much to help me learn linear algebra for my research. Thank you!!

[@drewsam6387]

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[@azariahivan3939]

@Drew Sam instablaster ;)

[@drewsam6387]

@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.

[@drewsam6387]

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[@azariahivan3939]

@Drew Sam Glad I could help :D

[@mahmudshamim1437]

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.

[@divyathangaraj9265]

Thank you so much for the beautiful ideas behind the circulant matrices. Geometric ideas help me to understand the concepts easily, thank you.

[@sandyjr5225]

Prof Strang, thank you so much for all your lectures.

[@adamodimattia]

The legend! Thank you!

[@georgesadler7830]

Professor Strang, thank you for explaining the Eigenvectors of Circulant Matrices ,Fourier Matrix and their impact on Machine Learning.

[@amesoeurs]

why do you just copy and paste the title on every single one of these videos?

[@exoticcoder5365]

@@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

[@Enerdzizer]

Interesting facts about normal matrices!

[@vaghawanojha8445]

at 51:00 did he mean e^(8*pi/3) which would be 1 instead of e^(6*pi/3)?

[@tapioms]

No! is in facto because e^(6 pi/3) =e^(4 pi/3)*e^(2 pi/3)=e^(4 pi/3+2 pi/3)

[@jerrymahone335]

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.