Lecture 31: Eigenvectors of Circulant Matrices: Fourier Matrix
Description
This lecture continues with constant-diagonal circulant matrices. Each lower diagonal continues on an upper diagonal to produce \(n\) equal entries. The eigenvectors are always the columns of the Fourier matrix and computing is fast.
Summary
Circulants \(C\) have \(n\) constant diagonals (completed cyclically).
Cyclic convolution with \(c_0, …, c_{n-1} =\) multiplication by \(C\)
Linear shift invariant: LSI for periodic problems
Eigenvectors of every \(C =\) columns of the Fourier matrix
Eigenvalues of \(C =\) (Fourier matrix)(column zero of \(C\))
Related section in textbook: IV.2
Instructor: Prof. Gilbert Strang
Problems for Lecture 31
From textbook Section IV.2
3. If \(\boldsymbol{c\ast d} = \boldsymbol{e}\), why is \((\sum c_i)(\sum d_i)=(\sum e_i)\)? Why was our check successful?
\((1+2+3)\,(5+0+4)=(\boldsymbol{6})\,(\boldsymbol{9})=\boldsymbol{54}=5+10+19+8+12\).
5. What are the eigenvalues of the 4 by 4 circulant \(C=I+P+P^2+P^3\)? Connect those eigenvalues to the discrete transform \(F\boldsymbol{c}\) for \(\boldsymbol{c}=(1,1,1,1)\). For which three real or complex numbers \(z\) is \(1+z+z^2+z^3=0\)?
