<s>
In	O
mathematics	O
,	O
a	O
bisymmetric	B-Algorithm
matrix	I-Algorithm
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
that	O
is	O
symmetric	O
about	O
both	O
of	O
its	O
main	O
diagonals	O
.	O
</s>
<s>
More	O
precisely	O
,	O
an	O
n×n	O
matrix	O
A	O
is	O
bisymmetric	O
if	O
it	O
satisfies	O
both	O
A	O
=	O
AT	O
and	O
AJ	O
=	O
JA	O
where	O
J	O
is	O
the	O
n×n	O
exchange	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Bisymmetric	O
matrices	O
are	O
both	O
symmetric	O
centrosymmetric	B-Algorithm
and	O
symmetric	O
persymmetric	B-Algorithm
.	O
</s>
<s>
The	O
product	O
of	O
two	O
bisymmetric	O
matrices	O
is	O
a	O
centrosymmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Real-valued	O
bisymmetric	O
matrices	O
are	O
precisely	O
those	O
symmetric	O
matrices	O
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>
If	O
A	O
is	O
a	O
real	O
bisymmetric	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
bisymmetric	O
.	O
</s>
