Lecture 12: Computing Eigenvalues and Singular Values
Description
Numerical linear algebra is the subject of this lecture and, in particular, how to compute eigenvalues and singular values. This includes discussion of the Hessenberg matrix, a square matrix that is almost (except for one extra diagonal) triangular.
Summary
\(QR\) method for eigenvalues: Reverse \(A = QR\) to \(A_1 = RQ\)
Then reverse \(A_1 = Q_1R_1\) to \(A_2 = R_1Q_1\): Include shifts
\(A\)’s become triangular with eigenvalues on the diagonal.
Krylov spaces and Krylov iterations
Related section in textbook: II.1
Instructor: Prof. Gilbert Strang
Problem for Lecture 12
From textbook Section II.1
These problems start with a bidiagonal \(n\) by \(n\) backward difference matrix \(D = I − S\). Two tridiagonal second difference matrices are \(DD^{\mathtt{T}}\) and \(A = −S + 2I −S^{\mathtt{T}}\). The shift \(S\) has one nonzero subdiagonal \(S_{i, i-1}=1\) for \(i=2,\ldots,n\). \(A\) has diagonals −1, 2, −1.
1. Show that \(DD^{\mathtt{T}}\) equals \(A\) except that \(1\neq 2\) in their (1, 1) entries. Similarly \(D^{\mathtt{T}} D = A\) except that \(1\neq 2\) in their \((n,n)\) entries.
