<s>
In	O
mathematics	O
,	O
a	O
pseudoreflection	B-Algorithm
is	O
an	O
invertible	O
linear	B-Architecture
transformation	I-Architecture
of	O
a	O
finite-dimensional	O
vector	O
space	O
such	O
that	O
it	O
is	O
not	O
the	O
identity	O
transformation	O
,	O
has	O
a	O
finite	O
(	O
multiplicative	O
)	O
order	O
,	O
and	O
fixes	O
a	O
hyperplane	O
.	O
</s>
<s>
The	O
concept	O
of	O
pseudoreflection	B-Algorithm
generalizes	O
the	O
concepts	O
of	O
reflection	B-Algorithm
and	O
complex	O
reflection	B-Algorithm
and	O
is	O
simply	O
called	O
reflection	B-Algorithm
by	O
some	O
mathematicians	O
.	O
</s>
<s>
Suppose	O
that	O
V	O
is	O
vector	O
space	O
over	O
a	O
field	O
K	O
,	O
whose	O
dimension	O
is	O
a	O
finite	O
number	O
n	O
.	O
A	O
pseudoreflection	B-Algorithm
is	O
an	O
invertible	O
linear	B-Architecture
transformation	I-Architecture
such	O
that	O
the	O
order	O
of	O
g	O
is	O
finite	O
and	O
the	O
fixed	O
subspace	O
of	O
all	O
vectors	O
in	O
V	O
fixed	O
by	O
g	O
has	O
dimension	O
n-1	O
.	O
</s>
<s>
A	O
pseudoreflection	B-Algorithm
g	O
has	O
an	O
eigenvalue	O
1	O
of	O
multiplicity	O
n-1	O
and	O
another	O
eigenvalue	O
r	O
of	O
multiplicity	O
1	O
.	O
</s>
<s>
A	O
diagonalizable	B-Algorithm
pseudoreflection	B-Algorithm
is	O
sometimes	O
called	O
a	O
semisimple	O
reflection	B-Algorithm
.	O
</s>
<s>
When	O
K	O
is	O
the	O
field	O
of	O
real	O
numbers	O
,	O
a	O
pseudoreflection	B-Algorithm
has	O
matrix	O
form	O
diag( 1	O
,	O
...	O
,	O
1	O
,	O
-1	O
)	O
.	O
</s>
<s>
A	O
pseudoreflection	B-Algorithm
with	O
such	O
matrix	O
form	O
is	O
called	O
a	O
real	O
reflection	B-Algorithm
.	O
</s>
<s>
If	O
the	O
space	O
on	O
which	O
this	O
transformation	O
acts	O
admits	O
a	O
symmetric	O
bilinear	O
form	O
so	O
that	O
orthogonality	B-Application
of	O
vectors	O
can	O
be	O
defined	O
,	O
then	O
the	O
transformation	O
is	O
a	O
true	O
reflection	B-Algorithm
.	O
</s>
<s>
When	O
K	O
is	O
the	O
field	O
of	O
complex	O
numbers	O
,	O
a	O
pseudoreflection	B-Algorithm
is	O
called	O
a	O
complex	O
reflection	B-Algorithm
,	O
which	O
can	O
be	O
represented	O
by	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
diag( 1	O
,	O
...	O
,	O
1	O
,	O
r	O
)	O
where	O
r	O
is	O
a	O
complex	O
root	O
of	O
unity	O
unequal	O
to	O
1	O
.	O
</s>
<s>
A	O
pseudoreflection	B-Algorithm
g	O
is	O
a	O
transvection	O
if	O
and	O
only	O
if	O
the	O
characteristic	O
p	O
of	O
the	O
field	O
K	O
is	O
positive	O
and	O
the	O
order	O
of	O
g	O
is	O
p	O
.	O
Transvections	O
are	O
useful	O
in	O
the	O
study	O
of	O
finite	O
geometries	O
and	O
the	O
classification	O
of	O
their	O
groups	O
of	O
motions	O
.	O
</s>
