Let

Determine P and Q as in Exercise 27, and compute Px and Qx. The figure shows that Qx is the reflection of x through the x₁x₂-plane.

A Householder reflection through the plane x3 = 0.

Data from in Exercise 27

Given u in Rⁿ with u^Tu = 1, let P = uu^T (an outer product) and Q = I - 2P. Justify statements,

(a) P² = P

(b) P^T = P

(c) Q² = 1

The transformation x ↦ Px is called a projection, and x ↦ Qx 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).

Transcribed Image Text:

u 0 0 and x = 1 1 53 3