Lecture 34: Distance Matrices, Procrustes Problem
Description
This lecture continues the review of distance matrices. Professor Strang then introduces the Procrustes problem, which looks for the orthogonal matrix that swings one set of vectors as nearly as possible onto a second set.
Summary
Distance problem: Find positions \(x\) from distances between them.
Necessary and sufficient: Distances satisfy triangle inequality.
Procrustes: Given \(n\) vectors \(x\) and \(n\) vectors \(y\).
Find the orthogonal matrix \(Q\) so that \(Qx\)’s are closest to \(y\)’s.
Related sections in textbook: IV.9 and IV.10
Instructor: Prof. Gilbert Strang
Problem for Lecture 34
From textbook Sections IV.9 and IV.10
1. Which orthogonal matrix \(Q\) minimizes \(||X-YQ||^2_F\)? Use the solution \(Q=UV^{\mathtt{T}} \) above and also minimize as a function of \(\theta\) (set the \(\theta\)-derivative to zero):
$$X = \left[\begin{matrix}1 & 2 \\ 2 &1\end{matrix}\right] \hspace{12pt} Y = \left[\begin{matrix}1 & 0\\ 0 & 1\end{matrix}\right] \hspace{12pt} Q = \left[\begin{matrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{matrix}\right] $$
