Householder Reflection

Definition 2.1.1. If and then the matrix of the form
 
is called the Householder matrix or Householder reflection and the vector is called the Householder vector.

Proposition 2.1.1. The Householder matrix H is symmetric and orthogonal. The Householder reflection reflects every vector in the hyperplane .

Proof. Indeed,

and

To prove the third part of the assertion, we choose on the hyperplane an orthogonal basis Hence, (i= 1:) and =1:. If

then







i.e., the vectors and have onto the hyperplane the same orthogonal projection

but projections onto the vector have opposite directions. Thus is the reflection of in the hyperplane . It is significant to note that the Householder matrix H depends only on the direction of Householder vektor and does not depend on the sign of the direction and length of .

Proposition 2.1.2. If and then vector , where H is the Householder matrix denoted by (1), has the same direction as , i.e., the Householder reflection H applied to the vector annihilates all but the first component of the vector .

Proof. Our aim is to determine for a nonzero vector the Householder vector so that Since

and then By choosing we obtain that



and



Choose so that in the latter representation of the coefficient of is zero, i.e.,





For this choice we have and

Example 2.1.1 Let Find the Householder vector and according to it the Householder transformation that annihilates the two last coordinates of the vector . By Proposition 2.1.1 we compute Choose the sign plus for coefficient of
and we obtain Find the Householder matrix H that depends only on direction of ,



Check,

Exercise 2..1.1.* Find the Householder matrix H such that , where

Let (=1:) be the Householder matrices. Consider the product of these matrices

where

and each has the form

The matrix Q can be written in the form
 
where W and Y are matrices. The answer to the question how to find representation (2) is given with the following proposition.

Proposition 2.1.3. Suppose is an orthogonal matrix with If where and then

where and and consequently, W₊,

Proof. Since



and

then and the assertion of the proposition holds.