<s>
In	O
mathematics	O
,	O
a	O
square	O
matrix	B-Architecture
is	O
said	O
to	O
be	O
diagonally	B-Algorithm
dominant	I-Algorithm
if	O
,	O
for	O
every	O
row	O
of	O
the	O
matrix	B-Architecture
,	O
the	O
magnitude	O
of	O
the	O
diagonal	O
entry	O
in	O
a	O
row	O
is	O
larger	O
than	O
or	O
equal	O
to	O
the	O
sum	O
of	O
the	O
magnitudes	O
of	O
all	O
the	O
other	O
(	O
non-diagonal	O
)	O
entries	O
in	O
that	O
row	O
.	O
</s>
<s>
This	O
definition	O
uses	O
a	O
weak	O
inequality	O
,	O
and	O
is	O
therefore	O
sometimes	O
called	O
weak	O
diagonal	B-Algorithm
dominance	I-Algorithm
.	O
</s>
<s>
If	O
a	O
strict	O
inequality	O
(	O
>	O
)	O
is	O
used	O
,	O
this	O
is	O
called	O
strict	O
diagonal	B-Algorithm
dominance	I-Algorithm
.	O
</s>
<s>
The	O
unqualified	O
term	O
diagonal	B-Algorithm
dominance	I-Algorithm
can	O
mean	O
both	O
strict	O
and	O
weak	O
diagonal	B-Algorithm
dominance	I-Algorithm
,	O
depending	O
on	O
the	O
context	O
.	O
</s>
<s>
It	O
is	O
therefore	O
sometimes	O
called	O
row	O
diagonal	B-Algorithm
dominance	I-Algorithm
.	O
</s>
<s>
If	O
one	O
changes	O
the	O
definition	O
to	O
sum	O
down	O
each	O
column	O
,	O
this	O
is	O
called	O
column	O
diagonal	B-Algorithm
dominance	I-Algorithm
.	O
</s>
<s>
Any	O
strictly	B-Algorithm
diagonally	I-Algorithm
dominant	I-Algorithm
matrix	B-Architecture
is	O
trivially	O
a	O
weakly	B-Algorithm
chained	I-Algorithm
diagonally	I-Algorithm
dominant	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Weakly	B-Algorithm
chained	I-Algorithm
diagonally	I-Algorithm
dominant	I-Algorithm
matrices	I-Algorithm
are	O
nonsingular	O
and	O
include	O
the	O
family	O
of	O
irreducibly	O
diagonally	B-Algorithm
dominant	I-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
These	O
are	O
irreducible	O
matrices	O
that	O
are	O
weakly	O
diagonally	B-Algorithm
dominant	I-Algorithm
,	O
but	O
strictly	B-Algorithm
diagonally	I-Algorithm
dominant	I-Algorithm
in	O
at	O
least	O
one	O
row	O
.	O
</s>
<s>
That	O
is	O
,	O
the	O
first	O
and	O
third	O
rows	O
fail	O
to	O
satisfy	O
the	O
diagonal	B-Algorithm
dominance	I-Algorithm
condition	O
.	O
</s>
<s>
A	O
strictly	B-Algorithm
diagonally	I-Algorithm
dominant	I-Algorithm
matrix	I-Algorithm
(	O
or	O
an	O
irreducibly	O
diagonally	B-Algorithm
dominant	I-Algorithm
matrix	I-Algorithm
)	O
is	O
non-singular	O
.	O
</s>
<s>
A	O
Hermitian	B-Algorithm
diagonally	B-Algorithm
dominant	I-Algorithm
matrix	I-Algorithm
with	O
real	O
non-negative	O
diagonal	O
entries	O
is	O
positive	B-Algorithm
semidefinite	I-Algorithm
.	O
</s>
<s>
If	O
the	O
symmetry	O
requirement	O
is	O
eliminated	O
,	O
such	O
a	O
matrix	B-Architecture
is	O
not	O
necessarily	O
positive	B-Algorithm
semidefinite	I-Algorithm
.	O
</s>
<s>
Similarly	O
,	O
a	O
Hermitian	B-Algorithm
strictly	B-Algorithm
diagonally	I-Algorithm
dominant	I-Algorithm
matrix	B-Architecture
with	O
real	O
positive	O
diagonal	O
entries	O
is	O
positive	B-Algorithm
definite	I-Algorithm
.	O
</s>
<s>
No	O
(	O
partial	O
)	O
pivoting	B-Algorithm
is	O
necessary	O
for	O
a	O
strictly	O
column	O
diagonally	B-Algorithm
dominant	I-Algorithm
matrix	I-Algorithm
when	O
performing	O
Gaussian	B-Algorithm
elimination	I-Algorithm
(	O
LU	O
factorization	O
)	O
.	O
</s>
<s>
The	O
Jacobi	B-Algorithm
and	O
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
methods	I-Algorithm
for	O
solving	O
a	O
linear	O
system	O
converge	O
if	O
the	O
matrix	B-Architecture
is	O
strictly	O
(	O
or	O
irreducibly	O
)	O
diagonally	B-Algorithm
dominant	I-Algorithm
.	O
</s>
<s>
Many	O
matrices	O
that	O
arise	O
in	O
finite	B-Application
element	I-Application
methods	I-Application
are	O
diagonally	B-Algorithm
dominant	I-Algorithm
.	O
</s>
<s>
A	O
slight	O
variation	O
on	O
the	O
idea	O
of	O
diagonal	B-Algorithm
dominance	I-Algorithm
is	O
used	O
to	O
prove	O
that	O
the	O
pairing	O
on	O
diagrams	O
without	O
loops	O
in	O
the	O
Temperley	B-Application
–	I-Application
Lieb	I-Application
algebra	I-Application
is	O
nondegenerate	O
.	O
</s>
<s>
For	O
a	O
matrix	B-Architecture
with	O
polynomial	O
entries	O
,	O
one	O
sensible	O
definition	O
of	O
diagonal	B-Algorithm
dominance	I-Algorithm
is	O
if	O
the	O
highest	O
power	O
of	O
appearing	O
in	O
each	O
row	O
appears	O
only	O
on	O
the	O
diagonal	O
.	O
</s>
<s>
(	O
The	O
evaluations	O
of	O
such	O
a	O
matrix	B-Architecture
at	O
large	O
values	O
of	O
are	O
diagonally	B-Algorithm
dominant	I-Algorithm
in	O
the	O
above	O
sense	O
.	O
)	O
</s>
