<s>
In	O
mathematics	O
,	O
a	O
hollow	B-Algorithm
matrix	I-Algorithm
may	O
refer	O
to	O
one	O
of	O
several	O
related	O
classes	O
of	O
matrix	B-Architecture
:	O
a	O
sparse	B-Algorithm
matrix	I-Algorithm
;	O
a	O
matrix	B-Architecture
with	O
a	O
large	O
block	O
of	O
zeroes	O
;	O
or	O
a	O
matrix	B-Architecture
with	O
diagonal	O
entries	O
all	O
zero	O
.	O
</s>
<s>
A	O
hollow	B-Algorithm
matrix	I-Algorithm
may	O
be	O
one	O
with	O
"	O
few	O
"	O
non-zero	O
entries	O
:	O
that	O
is	O
,	O
a	O
sparse	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
hollow	B-Algorithm
matrix	I-Algorithm
may	O
be	O
a	O
square	O
n×n	O
matrix	B-Architecture
with	O
an	O
r	O
×s	O
block	O
of	O
zeroes	O
where	O
r	O
+s	O
>	O
n	O
.	O
</s>
<s>
A	O
hollow	B-Algorithm
matrix	I-Algorithm
may	O
be	O
a	O
square	B-Algorithm
matrix	I-Algorithm
whose	O
diagonal	O
elements	O
are	O
all	O
equal	O
to	O
zero	O
.	O
</s>
<s>
That	O
is	O
,	O
an	O
n×n	O
matrix	B-Architecture
A	O
=	O
(	O
aij	O
)	O
is	O
hollow	O
if	O
aij	O
=	O
0	O
whenever	O
i	O
=	O
j	O
(	O
i.e.	O
</s>
<s>
The	O
most	O
obvious	O
example	O
is	O
the	O
real	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Other	O
examples	O
are	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
a	O
finite	O
simple	O
graph	O
,	O
and	O
a	O
distance	O
matrix	B-Architecture
or	O
Euclidean	B-Algorithm
distance	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
is	O
a	O
hollow	B-Algorithm
matrix	I-Algorithm
,	O
where	O
the	O
symbol	O
denotes	O
an	O
arbitrary	O
entry	O
.	O
</s>
<s>
is	O
a	O
hollow	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
trace	O
of	O
a	O
hollow	B-Algorithm
matrix	I-Algorithm
is	O
zero	O
.	O
</s>
<s>
If	O
A	O
represents	O
a	O
linear	B-Architecture
map	I-Architecture
with	O
respect	O
to	O
a	O
fixed	O
basis	O
,	O
then	O
it	O
maps	O
each	O
basis	O
vector	O
e	O
into	O
the	O
complement	O
of	O
the	O
span	O
of	O
e	O
.	O
That	O
is	O
,	O
where	O
.	O
</s>
<s>
The	O
Gershgorin	O
circle	O
theorem	O
shows	O
that	O
the	O
moduli	O
of	O
the	O
eigenvalues	O
of	O
a	O
hollow	B-Algorithm
matrix	I-Algorithm
are	O
less	O
or	O
equal	O
to	O
the	O
sum	O
of	O
the	O
moduli	O
of	O
the	O
non-diagonal	O
row	O
entries	O
.	O
</s>
