A Householder matrix, or an elementary reflector, has the form Q = 1- 2uuT where u is a unit vector. (See Exercise 13 in the Supplementary Exercises for Chapter 2.) Show that Q is an orthogonal matrix. (Elementary reflectors are often used in computer programs to produce a QR factorization of a matrix A. If A has linearly independent columns, then left- multiplication by a sequence of elementary reflectors can produce an upper triangular matrix.) Reference Exercise 13 in the Supplementary Exercises for Chapter 2: Given u in R" with u’u = 1, let P and (c). um" (an outer product) and Q = |- 2P. Justify statements (a), (b), a. P? = P b. PT = P c. Q? = I The transformation x> Px is called a projection, and x> Px is called a Householder reflection. Such reflections are used in computer programs to create multiple zeros in a vector (usually a column of a matrix).
A Householder matrix, or an elementary reflector, has the form Q = 1- 2uuT where u is a unit vector. (See Exercise 13 in the Supplementary Exercises for Chapter 2.) Show that Q is an orthogonal matrix. (Elementary reflectors are often used in computer programs to produce a QR factorization of a matrix A. If A has linearly independent columns, then left- multiplication by a sequence of elementary reflectors can produce an upper triangular matrix.) Reference Exercise 13 in the Supplementary Exercises for Chapter 2: Given u in R" with u’u = 1, let P and (c). um" (an outer product) and Q = |- 2P. Justify statements (a), (b), a. P? = P b. PT = P c. Q? = I The transformation x> Px is called a projection, and x> Px is called a Householder reflection. Such reflections are used in computer programs to create multiple zeros in a vector (usually a column of a matrix).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:A Householder matrix, or an elementary reflector, has the form Q = 1- 2uuT where u is a unit vector. (See Exercise 13 in the
Supplementary Exercises for Chapter 2.) Show that Q is an orthogonal matrix. (Elementary reflectors are often used in
computer programs to produce a QR factorization of a matrix A. If A has linearly independent columns, then left-
multiplication by a sequence of elementary reflectors can produce an upper triangular matrix.)
Reference Exercise 13 in the Supplementary Exercises for Chapter 2:
Given u in R" with u’u = 1, let P
and (c).
um" (an outer product) and Q = |- 2P. Justify statements (a), (b),
a. P? = P
b. PT = P
c. Q? = I
The transformation x> Px is called a projection, and x> Px is called a Householder reflection. Such
reflections are used in computer programs to create multiple zeros in a vector (usually a column of a matrix).
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