Lecture 12: Computing Eigenvalues and Singular Values

Course Info

Instructor

Prof. Gilbert Strang

Departments

Mathematics

As Taught In

Spring 2018

Level

Undergraduate

Topics

Learning Resource Types

Lecture Videos

Problem Sets

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Video Lectures

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

$Q R$ method for eigenvalues: Reverse $A = Q R$ to $A_{1} = R Q$
Then reverse $A_{1} = Q_{1} R_{1}$ to $A_{2} = R_{1} Q_{1}$ : 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 $D D^{T}$ and $A = - S + 2 I - S^{T}$ . The shift $S$ has one nonzero subdiagonal $S_{i, i - 1} = 1$ for $i = 2, \dots, n$ . $A$ has diagonals −1, 2, −1.

1. Show that $D D^{T}$ equals $A$ except that $1 \neq 2$ in their (1, 1) entries. Similarly $D^{T} D = A$ except that $1 \neq 2$ in their $(n, n)$ entries.