Lecture 30: Completing a Rank-One Matrix, Circulants!
Description
Professor Strang starts this lecture asking the question “Which matrices can be completed to have a rank of 1?” He then provides several examples. In the second part, he introduces convolution and cyclic convolution.
Summary
Which matrices can be completed to have rank = 1?
Perfect answer: No cycles in a certain graph
Cyclic permutation \(P\) and circulant matrices
\(c_0 I + c_1 P + c_2 P^2 + \cdots\)
Start of Fourier analysis for vectors
Related section in textbook: IV.8 and IV.2
Instructor: Prof. Gilbert Strang
Problems for Lecture 30
From textbook Section IV.8
3. For a connected graph with \(M\) edges and \(N\) nodes, what requirement on \(M\) and \(N\) comes from each of the words spanning tree?
From textbook Section IV.2
1. Find \(\boldsymbol{c\ast d}\) and \(\boldsymbol{c \circledast d}\) for \(\boldsymbol{c}=(2,1,3)\) and \(\boldsymbol{d}=(3,1,2)\).
