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!####[[' Prof Strang!', 'Instructor', 'POS'], ['eigenvectors of a circular matrix ', 'Mathematical_Related_Concept', 'POS'], ['Fourier Transform', 'Mathematical_Related_Concept', 'POS'], [' DCT', 'Mathematical_Related_Concept', 'POS'], ['convolution', 'Mathematical_Related_Concept', 'POS'], ['polynomial multiplication', 'Mathematical_Related_Concept', 'POS']]