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