Lecture 31: Eigenvectors of Circulant Matrices: Fourier Matrix

Course Info

Instructor

Prof. Gilbert Strang

Departments

Mathematics

As Taught In

Spring 2018

Level

Undergraduate

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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 $c_{0}, \dots, c_{n - 1} =$ 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 $c * d = e$ , why is $(\sum c_{i}) (\sum d_{i}) = (\sum e_{i})$ ? Why was our check successful?
$(1 + 2 + 3) (5 + 0 + 4) = (6) (9) = 54 = 5 + 10 + 19 + 8 + 12$ .

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 $F c$ for $c = (1, 1, 1, 1)$ . For which three real or complex numbers $z$ is $1 + z + z^{2} + z^{3} = 0$ ?