Solution Found!
A Householder matrix, or an elementary reflector, has the
Chapter , Problem 7E(choose chapter or problem)
Problem 7E
A Householder matrix, or an elementary reflector, has the form Q = I – 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 (an outer product) and Q = I – 2P . Justify statements (a), (b), and (c).
The transformation is called a projection, and 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).
Questions & Answers
QUESTION:
Problem 7E
A Householder matrix, or an elementary reflector, has the form Q = I – 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 (an outer product) and Q = I – 2P . Justify statements (a), (b), and (c).
The transformation is called a projection, and 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).
ANSWER:
Solution 7E
Step 1 of 2
The elementary reflector is defined as,
, where
The objective is to show that Q is an orthogonal matrix, that is
Apply transpose to
