<s>
In	O
mathematics	O
,	O
especially	O
in	O
linear	B-Language
algebra	I-Language
and	O
matrix	B-Architecture
theory	I-Architecture
,	O
a	O
centrosymmetric	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	B-Architecture
which	O
is	O
symmetric	B-Algorithm
about	O
its	O
center	O
.	O
</s>
<s>
If	O
J	O
denotes	O
the	O
n×n	O
exchange	B-Algorithm
matrix	I-Algorithm
with	O
1	O
on	O
the	O
antidiagonal	B-Algorithm
and	O
0	O
elsewhere	O
(	O
that	O
is	O
,	O
Ji	O
,	O
n	O
+	O
1	O
−	O
i	O
=	O
1	O
;	O
Ji	O
,	O
j	O
=	O
0	O
if	O
j	O
≠	O
n	O
+1−	O
i	O
)	O
,	O
then	O
a	O
matrix	B-Architecture
A	O
is	O
centrosymmetric	O
if	O
and	O
only	O
if	O
AJ	O
=	O
JA	O
.	O
</s>
<s>
Symmetric	B-Algorithm
Toeplitz	B-Algorithm
matrices	I-Algorithm
are	O
centrosymmetric	O
.	O
</s>
<s>
If	O
A	O
and	O
B	O
are	O
centrosymmetric	O
matrices	O
over	O
a	O
field	O
F	O
,	O
then	O
so	O
are	O
A	O
+	O
B	O
and	O
cA	O
for	O
any	O
c	O
in	O
F	O
.	O
Moreover	O
,	O
the	O
matrix	B-Architecture
product	O
AB	O
is	O
centrosymmetric	O
,	O
since	O
JAB	O
=	O
AJB	O
=	O
ABJ	O
.	O
</s>
<s>
Since	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
is	O
also	O
centrosymmetric	O
,	O
it	O
follows	O
that	O
the	O
set	O
of	O
n×n	O
centrosymmetric	O
matrices	O
over	O
F	O
is	O
a	O
subalgebra	O
of	O
the	O
associative	O
algebra	O
of	O
all	O
n×n	O
matrices	O
.	O
</s>
<s>
If	O
A	O
is	O
a	O
centrosymmetric	B-Algorithm
matrix	I-Algorithm
with	O
an	O
m-dimensional	O
eigenbasis	O
,	O
then	O
its	O
m	O
eigenvectors	O
can	O
each	O
be	O
chosen	O
so	O
that	O
they	O
satisfy	O
either	O
x	O
=	O
Jx	O
or	O
x	O
=	O
−Jx	O
where	O
J	O
is	O
the	O
exchange	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
If	O
A	O
is	O
a	O
centrosymmetric	B-Algorithm
matrix	I-Algorithm
with	O
distinct	O
eigenvalues	O
,	O
then	O
the	O
matrices	O
that	O
commute	O
with	O
A	O
must	O
be	O
centrosymmetric	O
.	O
</s>
<s>
The	O
maximum	O
number	O
of	O
unique	O
elements	O
in	O
a	O
m	O
×	O
m	O
centrosymmetric	B-Algorithm
matrix	I-Algorithm
is	O
.	O
</s>
<s>
An	O
n×n	O
matrix	B-Architecture
A	O
is	O
said	O
to	O
be	O
skew-centrosymmetric	O
if	O
its	O
entries	O
satisfy	O
Ai	O
,	O
j	O
=	O
−An−i+1	O
,	O
n−j+1	O
for	O
i	O
,	O
j	O
∊	O
{	O
1	O
,	O
...	O
,	O
n}	O
.	O
</s>
<s>
Equivalently	O
,	O
A	O
is	O
skew-centrosymmetric	O
if	O
AJ	O
=	O
−JA	O
,	O
where	O
J	O
is	O
the	O
exchange	B-Algorithm
matrix	I-Algorithm
defined	O
above	O
.	O
</s>
<s>
The	O
centrosymmetric	O
relation	O
AJ	O
=	O
JA	O
lends	O
itself	O
to	O
a	O
natural	O
generalization	O
,	O
where	O
J	O
is	O
replaced	O
with	O
an	O
involutory	B-Algorithm
matrix	I-Algorithm
K	O
(	O
i.e.	O
,	O
K2	O
=	O
I	O
)	O
or	O
,	O
more	O
generally	O
,	O
a	O
matrix	B-Architecture
K	O
satisfying	O
Km	O
=	O
I	O
for	O
an	O
integer	O
m	O
>	O
1	O
.	O
</s>
<s>
The	O
inverse	O
problem	O
for	O
the	O
commutation	O
relation	O
of	O
identifying	O
all	O
involutory	O
K	O
that	O
commute	O
with	O
a	O
fixed	O
matrix	B-Architecture
A	O
has	O
also	O
been	O
studied	O
.	O
</s>
<s>
Symmetric	B-Algorithm
centrosymmetric	O
matrices	O
are	O
sometimes	O
called	O
bisymmetric	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
When	O
the	O
ground	O
field	O
is	O
the	O
field	O
of	O
real	O
numbers	O
,	O
it	O
has	O
been	O
shown	O
that	O
bisymmetric	B-Algorithm
matrices	I-Algorithm
are	O
precisely	O
those	O
symmetric	B-Algorithm
matrices	I-Algorithm
whose	O
eigenvalues	O
remain	O
the	O
same	O
aside	O
from	O
possible	O
sign	O
changes	O
following	O
pre	O
-	O
or	O
post-multiplication	O
by	O
the	O
exchange	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
similar	O
result	O
holds	O
for	O
Hermitian	B-Algorithm
centrosymmetric	O
and	O
skew-centrosymmetric	O
matrices	O
.	O
</s>
