Page 260 - Advanced Linear Algebra
P. 260
244 Advanced Linear Algebra
Definition For a nonzero #= , the unique operator / # for which
/# ~ c#Á /$ ~ $ for all $ º#»
#
#
is called a reflection or a Householder transformation .
It is easy to verify that
º%Á #»
/% ~ % c #
#
º#Á #»
Moreover, / % ~c% for % £ if and only if %~ # for some - and so
#
we can uniquely identify by the behavior of the reflection on .
=
#
If / is a reflection and if we extend to an ordered orthonormal basis for ,
8
=
# #
is the matrix obtained from the identity matrix by replacing the upper
then ´/ µ # 8
left entry by c ,
v c y
x {
´/ µ ~ x # 8 {
Æ
w z
Thus, a reflection is both unitary and Hermitian, that is,
i
/~ / c
# # ~ / #
Given two nonzero vectors of equal length, there is precisely one reflection that
interchanges these vectors.
Theorem 10.16 Let #Á $ = be distinct nonzero vectors of equal length. Then
/ # #c$ is the unique reflection sending to and to .
$
#
$
$
#
Proof. If )) ~ ) ) , then ²#c$³ ²#b$³ and so
/ ² #c$ # c $ ³ ~ $ c #
/ ² #c$ # b $ ³ ~ # b $
from which it follows that / ² #c$ # ³ ~ $ and / ² #c$ $ ³ ~ # . As to uniqueness,
suppose / is a reflection for which / % % ² # ³ ~ $ . Since / % c ~ / % , we have
/²$³ ~ # and so
%
/²# c $³ ~ c²# c $³
%
.
which implies that /~ / # % c $
Reflections can be used to characterize unitary operators.
Theorem 10.17 Let = be a finite-dimensional inner product space. The
B
following are equivalent for an operator ²= ³ :
