<s>
In	O
mathematics	O
,	O
a	O
conference	B-Algorithm
matrix	I-Algorithm
(	O
also	O
called	O
a	O
C-matrix	O
)	O
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
C	O
with	O
0	O
on	O
the	O
diagonal	O
and	O
+1	O
and	O
1	O
off	O
the	O
diagonal	O
,	O
such	O
that	O
CTC	O
is	O
a	O
multiple	O
of	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
I	O
.	O
</s>
<s>
Thus	O
,	O
if	O
the	O
matrix	B-Architecture
has	O
order	O
n	O
,	O
CTC	O
=(	O
n1	O
)	O
I	O
.	O
</s>
<s>
Belevitch	O
was	O
interested	O
in	O
constructing	O
ideal	O
telephone	O
conference	O
networks	O
from	O
ideal	O
transformers	B-Algorithm
and	O
discovered	O
that	O
such	O
networks	O
were	O
represented	O
by	O
conference	O
matrices	O
,	O
hence	O
the	O
name	O
.	O
</s>
<s>
For	O
n1	O
,	O
there	O
are	O
two	O
kinds	O
of	O
conference	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
(	O
These	O
operations	O
do	O
not	O
change	O
whether	O
a	O
matrix	B-Architecture
is	O
a	O
conference	B-Algorithm
matrix	I-Algorithm
.	O
)	O
</s>
<s>
Thus	O
,	O
a	O
normalized	O
conference	B-Algorithm
matrix	I-Algorithm
has	O
all	O
1	O
's	O
in	O
its	O
first	O
row	O
and	O
column	O
,	O
except	O
for	O
a	O
0	O
in	O
the	O
top	O
left	O
corner	O
,	O
and	O
is	O
0	O
on	O
the	O
diagonal	O
.	O
</s>
<s>
Let	O
S	O
be	O
the	O
matrix	B-Architecture
that	O
remains	O
when	O
the	O
first	O
row	O
and	O
column	O
of	O
C	O
are	O
removed	O
.	O
</s>
<s>
Then	O
either	O
n	O
is	O
evenly	O
even	O
(	O
a	O
multiple	O
of	O
4	O
)	O
,	O
and	O
S	O
is	O
antisymmetric	B-Algorithm
(	O
as	O
is	O
the	O
normalized	O
C	O
if	O
its	O
first	O
row	O
is	O
negated	O
)	O
,	O
or	O
n	O
is	O
oddly	O
even	O
(	O
congruent	O
to	O
2	O
modulo	O
4	O
)	O
and	O
S	O
is	O
symmetric	B-Algorithm
(	O
as	O
is	O
the	O
normalized	O
C	O
)	O
.	O
</s>
<s>
If	O
C	O
is	O
a	O
symmetric	B-Algorithm
conference	B-Algorithm
matrix	I-Algorithm
of	O
order	O
n1	O
,	O
then	O
not	O
only	O
must	O
n	O
be	O
congruent	O
to	O
2	O
(	O
mod	O
4	O
)	O
but	O
also	O
n	O
1	O
must	O
be	O
a	O
sum	O
of	O
two	O
square	O
integers	O
;	O
there	O
is	O
a	O
clever	O
proof	O
by	O
elementary	O
matrix	B-Architecture
theory	I-Architecture
in	O
van	O
Lint	O
and	O
Seidel	O
.	O
</s>
<s>
Given	O
a	O
symmetric	B-Algorithm
conference	B-Algorithm
matrix	I-Algorithm
,	O
the	O
matrix	B-Architecture
S	O
can	O
be	O
viewed	O
as	O
the	O
Seidel	O
adjacency	O
matrix	B-Architecture
of	O
a	O
graph	O
.	O
</s>
<s>
This	O
graph	O
is	O
strongly	O
regular	O
of	O
the	O
type	O
called	O
(	O
after	O
the	O
matrix	B-Architecture
)	O
a	O
conference	O
graph	O
.	O
</s>
<s>
The	O
existence	O
of	O
conference	O
matrices	O
of	O
orders	O
n	O
allowed	O
by	O
the	O
above	O
restrictions	O
is	O
known	O
only	O
for	O
some	O
values	O
of	O
n	O
.	O
For	O
instance	O
,	O
if	O
n	O
=	O
q	O
+	O
1	O
where	O
q	O
is	O
a	O
prime	O
power	O
congruent	O
to	O
1	O
(	O
mod	O
4	O
)	O
,	O
then	O
the	O
Paley	O
graphs	O
provide	O
examples	O
of	O
symmetric	B-Algorithm
conference	O
matrices	O
of	O
order	O
n	O
,	O
by	O
taking	O
S	O
to	O
be	O
the	O
Seidel	O
matrix	B-Architecture
of	O
the	O
Paley	O
graph	O
.	O
</s>
<s>
The	O
first	O
few	O
possible	O
orders	O
of	O
a	O
symmetric	B-Algorithm
conference	B-Algorithm
matrix	I-Algorithm
are	O
n	O
=	O
2	O
,	O
6	O
,	O
10	O
,	O
14	O
,	O
18	O
,	O
(	O
not	O
22	O
,	O
since	O
21	O
is	O
not	O
a	O
sum	O
of	O
two	O
squares	O
)	O
,	O
26	O
,	O
30	O
,	O
(	O
not	O
34	O
since	O
33	O
is	O
not	O
a	O
sum	O
of	O
two	O
squares	O
)	O
,	O
38	O
,	O
42	O
,	O
46	O
,	O
50	O
,	O
54	O
,	O
(	O
not	O
58	O
)	O
,	O
62	O
;	O
for	O
every	O
one	O
of	O
these	O
,	O
it	O
is	O
known	O
that	O
a	O
symmetric	B-Algorithm
conference	B-Algorithm
matrix	I-Algorithm
of	O
that	O
order	O
exists	O
.	O
</s>
<s>
Antisymmetric	B-Algorithm
matrices	I-Algorithm
can	O
also	O
be	O
produced	O
by	O
the	O
Paley	O
construction	O
.	O
</s>
<s>
Then	O
there	O
is	O
a	O
Paley	O
digraph	O
of	O
order	O
q	O
which	O
leads	O
to	O
an	O
antisymmetric	B-Algorithm
conference	B-Algorithm
matrix	I-Algorithm
of	O
order	O
n	O
=	O
q	O
+	O
1	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
is	O
obtained	O
by	O
taking	O
for	O
S	O
the	O
q	O
q	O
matrix	B-Architecture
that	O
has	O
a	O
+1	O
in	O
position	O
(	O
i	O
,	O
j	O
)	O
and	O
1	O
in	O
position	O
(	O
j	O
,	O
i	O
)	O
if	O
there	O
is	O
an	O
arc	O
of	O
the	O
digraph	O
from	O
i	O
to	O
j	O
,	O
and	O
zero	O
diagonal	O
.	O
</s>
<s>
Then	O
C	O
constructed	O
as	O
above	O
from	O
S	O
,	O
but	O
with	O
the	O
first	O
row	O
all	O
negative	O
,	O
is	O
an	O
antisymmetric	B-Algorithm
conference	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
This	O
construction	O
solves	O
only	O
a	O
small	O
part	O
of	O
the	O
problem	O
of	O
deciding	O
for	O
which	O
evenly	O
even	O
numbers	O
n	O
there	O
exist	O
antisymmetric	B-Algorithm
conference	O
matrices	O
of	O
order	O
n	O
.	O
</s>
<s>
W(n,w )	O
is	O
said	O
to	O
be	O
of	O
weight	O
w>0	O
and	O
order	O
n	O
if	O
it	O
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
of	O
size	O
n	O
with	O
entries	O
from	O
{−1	O
,	O
0	O
,	O
+1}	O
satisfying	O
W	O
Wt	O
=	O
w	O
I	O
.	O
</s>
<s>
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
and	O
at	O
most	O
one	O
zero	O
in	O
each	O
row	O
.	O
</s>
<s>
The	O
network	O
must	O
contain	O
ideal	O
transformers	B-Algorithm
only	O
and	O
no	O
resistances	O
.	O
</s>
<s>
An	O
n-port	O
ideal	O
conference	O
network	O
exists	O
if	O
and	O
only	O
if	O
there	O
exists	O
a	O
conference	B-Algorithm
matrix	I-Algorithm
of	O
order	O
n	O
.	O
For	O
instance	O
,	O
a	O
3-port	O
conference	O
network	O
can	O
be	O
constructed	O
with	O
the	O
well-known	O
hybrid	O
transformer	B-Algorithm
circuit	O
used	O
for	O
2-wire	O
to	O
4-wire	O
conversion	O
in	O
telephone	O
handsets	O
and	O
line	O
repeaters	O
.	O
</s>
<s>
However	O
,	O
there	O
is	O
no	O
order	O
3	O
conference	B-Algorithm
matrix	I-Algorithm
and	O
this	O
circuit	O
does	O
not	O
produce	O
an	O
ideal	O
conference	O
network	O
.	O
</s>
<s>
As	O
mentioned	O
above	O
,	O
a	O
necessary	O
condition	O
for	O
a	O
conference	B-Algorithm
matrix	I-Algorithm
to	O
exist	O
is	O
that	O
n1	O
must	O
be	O
the	O
sum	O
of	O
two	O
squares	O
.	O
</s>
