<s>
In	O
mathematics	O
,	O
an	O
involutory	B-Algorithm
matrix	I-Algorithm
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
that	O
is	O
its	O
own	O
inverse	O
.	O
</s>
<s>
That	O
is	O
,	O
multiplication	O
by	O
the	O
matrix	O
A	O
is	O
an	O
involution	B-Algorithm
if	O
and	O
only	O
if	O
A2	O
=	O
I	O
,	O
where	O
I	O
is	O
the	O
n×n	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Involutory	O
matrices	O
are	O
all	O
square	O
roots	O
of	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
special	O
case	O
of	O
another	O
class	O
of	O
elementary	O
matrix	O
,	O
that	O
which	O
represents	O
multiplication	O
of	O
a	O
row	O
or	O
column	O
by	O
−1	O
,	O
is	O
also	O
involutory	O
;	O
it	O
is	O
in	O
fact	O
a	O
trivial	O
example	O
of	O
a	O
signature	B-Algorithm
matrix	I-Algorithm
,	O
all	O
of	O
which	O
are	O
involutory	O
.	O
</s>
<s>
I	O
is	O
the	O
3×3	O
identity	B-Algorithm
matrix	I-Algorithm
(	O
which	O
is	O
trivially	O
involutory	O
)	O
;	O
</s>
<s>
R	O
is	O
the	O
3×3	O
identity	B-Algorithm
matrix	I-Algorithm
with	O
a	O
pair	O
of	O
interchanged	O
rows	O
;	O
</s>
<s>
S	O
is	O
a	O
signature	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
An	O
involutory	B-Algorithm
matrix	I-Algorithm
which	O
is	O
also	O
symmetric	B-Algorithm
is	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
,	O
and	O
thus	O
represents	O
an	O
isometry	O
(	O
a	O
linear	B-Architecture
transformation	I-Architecture
which	O
preserves	O
Euclidean	O
distance	O
)	O
.	O
</s>
<s>
Conversely	O
every	O
orthogonal	O
involutory	B-Algorithm
matrix	I-Algorithm
is	O
symmetric	B-Algorithm
.	O
</s>
<s>
As	O
a	O
special	O
case	O
of	O
this	O
,	O
every	O
reflection	B-Algorithm
and	O
180°	O
rotation	B-Algorithm
matrix	I-Algorithm
is	O
involutory	O
.	O
</s>
<s>
An	O
involution	B-Algorithm
is	O
non-defective	O
,	O
and	O
each	O
eigenvalue	O
equals	O
,	O
so	O
an	O
involution	B-Algorithm
diagonalizes	B-Algorithm
to	O
a	O
signature	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
normal	B-Algorithm
involution	B-Algorithm
is	O
Hermitian	B-Algorithm
(	O
complex	O
)	O
or	O
symmetric	B-Algorithm
(	O
real	O
)	O
and	O
also	O
unitary	B-Algorithm
(	O
complex	O
)	O
or	O
orthogonal	O
(	O
real	O
)	O
.	O
</s>
<s>
The	O
determinant	O
of	O
an	O
involutory	B-Algorithm
matrix	I-Algorithm
over	O
any	O
field	O
is	O
±1	O
.	O
</s>
<s>
This	O
relation	O
gives	O
a	O
bijection	B-Algorithm
between	O
involutory	O
matrices	O
and	O
idempotent	O
matrices	O
.	O
</s>
<s>
These	O
two	O
operators	O
form	O
the	O
symmetric	B-Algorithm
and	O
antisymmetric	O
projections	O
of	O
a	O
vector	O
with	O
respect	O
to	O
the	O
involution	B-Algorithm
A	O
,	O
in	O
the	O
sense	O
that	O
,	O
or	O
.	O
</s>
<s>
The	O
same	O
construct	O
applies	O
to	O
any	O
involutory	B-Algorithm
function	I-Algorithm
,	O
such	O
as	O
the	O
complex	O
conjugate	O
(	O
real	O
and	O
imaginary	O
parts	O
)	O
,	O
transpose	O
(	O
symmetric	B-Algorithm
and	O
antisymetric	O
matrices	O
)	O
,	O
and	O
Hermitian	B-Algorithm
adjoint	O
(	O
Hermitian	B-Algorithm
and	O
skew-Hermitian	O
matrices	O
)	O
.	O
</s>
<s>
If	O
A	O
is	O
an	O
involutory	B-Algorithm
matrix	I-Algorithm
in	O
M(n, R )	O
,	O
a	O
matrix	O
algebra	O
over	O
the	O
real	O
numbers	O
,	O
then	O
the	O
subalgebra	O
{	O
xI	O
+	O
yA	O
:	O
x	O
,	O
y	O
∈	O
R}	O
generated	O
by	O
A	O
is	O
isomorphic	O
to	O
the	O
split-complex	O
numbers	O
.	O
</s>
<s>
If	O
A	O
is	O
an	O
involutory	B-Algorithm
matrix	I-Algorithm
then	O
every	O
integer	O
power	O
of	O
A	O
is	O
involutory	O
.	O
</s>
