<s>
In	O
linear	B-Language
algebra	I-Language
,	O
a	O
Householder	B-Algorithm
transformation	I-Algorithm
(	O
also	O
known	O
as	O
a	O
Householder	B-Algorithm
reflection	I-Algorithm
or	O
elementary	B-Algorithm
reflector	I-Algorithm
)	O
is	O
a	O
linear	B-Architecture
transformation	I-Architecture
that	O
describes	O
a	O
reflection	B-Algorithm
about	O
a	O
plane	O
or	O
hyperplane	O
containing	O
the	O
origin	O
.	O
</s>
<s>
The	O
Householder	B-Algorithm
transformation	I-Algorithm
was	O
used	O
in	O
a	O
1958	O
paper	O
by	O
Alston	O
Scott	O
Householder	O
.	O
</s>
<s>
The	O
reflection	B-Algorithm
hyperplane	O
can	O
be	O
defined	O
by	O
its	O
normal	O
vector	O
,	O
a	O
unit	O
vector	O
(	O
a	O
vector	O
with	O
length	O
)	O
that	O
is	O
orthogonal	O
to	O
the	O
hyperplane	O
.	O
</s>
<s>
The	O
reflection	B-Algorithm
of	O
a	O
point	O
about	O
this	O
hyperplane	O
is	O
the	O
linear	B-Architecture
transformation	I-Architecture
:	O
</s>
<s>
where	O
is	O
given	O
as	O
a	O
column	O
unit	O
vector	O
with	O
Hermitian	B-Algorithm
transpose	I-Algorithm
.	O
</s>
<s>
is	O
known	O
as	O
the	O
Householder	B-Algorithm
matrix	I-Algorithm
,	O
where	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
Householder	B-Algorithm
matrix	I-Algorithm
has	O
the	O
following	O
properties	O
:	O
</s>
<s>
it	O
is	O
Hermitian	B-Algorithm
:	O
,	O
</s>
<s>
it	O
is	O
unitary	B-Algorithm
:	O
,	O
</s>
<s>
hence	O
it	O
is	O
involutory	B-Algorithm
:	O
.	O
</s>
<s>
A	O
Householder	B-Algorithm
matrix	I-Algorithm
has	O
eigenvalues	O
.	O
</s>
<s>
To	O
see	O
this	O
,	O
notice	O
that	O
if	O
is	O
orthogonal	O
to	O
the	O
vector	O
which	O
was	O
used	O
to	O
create	O
the	O
reflector	B-Algorithm
,	O
then	O
,	O
i.e.	O
,	O
is	O
an	O
eigenvalue	O
of	O
multiplicity	O
,	O
since	O
there	O
are	O
independent	O
vectors	O
orthogonal	O
to	O
.	O
</s>
<s>
The	O
determinant	O
of	O
a	O
Householder	O
reflector	B-Algorithm
is	O
,	O
since	O
the	O
determinant	O
of	O
a	O
matrix	O
is	O
the	O
product	O
of	O
its	O
eigenvalues	O
,	O
in	O
this	O
case	O
one	O
of	O
which	O
is	O
with	O
the	O
remainder	O
being	O
(	O
as	O
in	O
the	O
previous	O
point	O
)	O
.	O
</s>
<s>
In	O
geometric	O
optics	O
,	O
specular	O
reflection	B-Algorithm
can	O
be	O
expressed	O
in	O
terms	O
of	O
the	O
Householder	B-Algorithm
matrix	I-Algorithm
(	O
see	O
)	O
.	O
</s>
<s>
Householder	B-Algorithm
transformations	I-Algorithm
are	O
widely	O
used	O
in	O
numerical	O
linear	B-Language
algebra	I-Language
,	O
for	O
example	O
,	O
to	O
annihilate	O
the	O
entries	O
below	O
the	O
main	O
diagonal	O
of	O
a	O
matrix	O
,	O
to	O
perform	O
QR	O
decompositions	O
and	O
in	O
the	O
first	O
step	O
of	O
the	O
QR	O
algorithm	O
.	O
</s>
<s>
They	O
are	O
also	O
widely	O
used	O
for	O
transforming	O
to	O
a	O
Hessenberg	B-Algorithm
form	I-Algorithm
.	O
</s>
<s>
For	O
symmetric	O
or	O
Hermitian	B-Algorithm
matrices	I-Algorithm
,	O
the	O
symmetry	O
can	O
be	O
preserved	O
,	O
resulting	O
in	O
tridiagonalization	O
.	O
</s>
<s>
Householder	B-Algorithm
reflections	I-Algorithm
can	O
be	O
used	O
to	O
calculate	O
QR	O
decompositions	O
by	O
reflecting	O
first	O
one	O
column	O
of	O
a	O
matrix	O
onto	O
a	O
multiple	O
of	O
a	O
standard	O
basis	O
vector	O
,	O
calculating	O
the	O
transformation	O
matrix	O
,	O
multiplying	O
it	O
with	O
the	O
original	O
matrix	O
and	O
then	O
recursing	O
down	O
the	O
minors	O
of	O
that	O
product	O
.	O
</s>
<s>
In	O
the	O
first	O
step	O
,	O
to	O
form	O
the	O
Householder	B-Algorithm
matrix	I-Algorithm
in	O
each	O
step	O
we	O
need	O
to	O
determine	O
and	O
,	O
which	O
are	O
:	O
</s>
<s>
Continuing	O
in	O
this	O
manner	O
,	O
the	O
tridiagonal	B-Algorithm
and	O
symmetric	O
matrix	O
is	O
formed	O
.	O
</s>
<s>
In	O
this	O
example	O
,	O
also	O
from	O
Burden	O
and	O
Faires	O
,	O
the	O
given	O
matrix	O
is	O
transformed	O
to	O
the	O
similar	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
A3	O
by	O
using	O
the	O
Householder	O
method	O
.	O
</s>
<s>
The	O
first	O
Householder	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
As	O
we	O
can	O
see	O
,	O
the	O
final	O
result	O
is	O
a	O
tridiagonal	B-Algorithm
symmetric	O
matrix	O
which	O
is	O
similar	O
to	O
the	O
original	O
one	O
.	O
</s>
<s>
The	O
Householder	B-Algorithm
transformation	I-Algorithm
is	O
a	O
reflection	B-Algorithm
about	O
a	O
hyperplane	O
with	O
unit	O
normal	O
vector	O
,	O
as	O
stated	O
earlier	O
.	O
</s>
<s>
An	O
-by	O
-	O
unitary	B-Algorithm
transformation	I-Algorithm
satisfies	O
.	O
</s>
<s>
Taking	O
the	O
determinant	O
(	O
-th	O
power	O
of	O
the	O
geometric	O
mean	O
)	O
and	O
trace	O
(	O
proportional	O
to	O
arithmetic	O
mean	O
)	O
of	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
reveals	O
that	O
its	O
eigenvalues	O
have	O
unit	O
modulus	O
.	O
</s>
<s>
For	O
the	O
case	O
of	O
real	O
valued	O
unitary	B-Algorithm
matrices	I-Algorithm
we	O
obtain	O
orthogonal	B-Algorithm
matrices	I-Algorithm
,	O
.	O
</s>
<s>
It	O
follows	O
rather	O
readily	O
(	O
see	O
orthogonal	B-Algorithm
matrix	I-Algorithm
)	O
that	O
any	O
orthogonal	B-Algorithm
matrix	I-Algorithm
can	O
be	O
decomposed	O
into	O
a	O
product	O
of	O
2	O
by	O
2	O
rotations	O
,	O
called	O
Givens	O
Rotations	O
,	O
and	O
Householder	B-Algorithm
reflections	I-Algorithm
.	O
</s>
<s>
This	O
is	O
appealing	O
intuitively	O
since	O
multiplication	O
of	O
a	O
vector	O
by	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
preserves	O
the	O
length	O
of	O
that	O
vector	O
,	O
and	O
rotations	O
and	O
reflections	O
exhaust	O
the	O
set	O
of	O
(	O
real	O
valued	O
)	O
geometric	O
operations	O
that	O
render	O
invariant	O
a	O
vector	O
's	O
length	O
.	O
</s>
<s>
The	O
Householder	B-Algorithm
transformation	I-Algorithm
was	O
shown	O
to	O
have	O
a	O
one-to-one	O
relationship	O
with	O
the	O
canonical	O
coset	O
decomposition	O
of	O
unitary	B-Algorithm
matrices	I-Algorithm
defined	O
in	O
group	O
theory	O
,	O
which	O
can	O
be	O
used	O
to	O
parametrize	O
unitary	B-Algorithm
operators	O
in	O
a	O
very	O
efficient	O
manner	O
.	O
</s>
<s>
Finally	O
we	O
note	O
that	O
a	O
single	O
Householder	B-Algorithm
transform	I-Algorithm
,	O
unlike	O
a	O
solitary	O
Givens	O
transform	O
,	O
can	O
act	O
on	O
all	O
columns	O
of	O
a	O
matrix	O
,	O
and	O
as	O
such	O
exhibits	O
the	O
lowest	O
computational	O
cost	O
for	O
QR	O
decomposition	O
and	O
tridiagonalization	O
.	O
</s>
