Lecture 17: Rapidly Decreasing Singular Values
Description
Professor Alex Townsend gives this guest lecture answering the question “Why are there so many low rank matrices that appear in computational math?” Working effectively with low rank matrices is critical in image compression applications.
Summary
Professor Alex Townsend’s lecture
Why do so many matrices have low effective rank?
Sylvester test for rapid decay of singular values
Image compression: Rank \(k\) needs only \(2kn\) numbers.
Flags give many examples / diagonal lines give high rank.
Related section in textbook: III.3
Instructor: Prof. Alex Townsend
Problems for Lecture 17
From textbook Section III.3
2. Show that the evil Hilbert matrix \(H\) passes the Sylvester test \(AH-HB=C\)
$$H_{ij}=\dfrac{1}{i+j-1}\quad A=\dfrac{1}{2}\hbox{diag}\,(1, 3,\ldots,2n-1)\quad B= -A\quad C=\textbf{ones}(n)$$
6. If an invertible matrix \(X\) satisfies the Sylvester equation \(AX-XB=C\), find a Sylvester equation for \(X^{-1}\).
