<s>
In	O
mathematics	O
,	O
particularly	O
matrix	B-Architecture
theory	I-Architecture
and	O
combinatorics	O
,	O
a	O
Pascal	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	B-Architecture
(	O
possibly	O
infinite	O
)	O
containing	O
the	O
binomial	O
coefficients	O
as	O
its	O
elements	O
.	O
</s>
<s>
It	O
is	O
thus	O
an	O
encoding	O
of	O
Pascal	O
's	O
triangle	O
in	O
matrix	B-Architecture
form	O
.	O
</s>
<s>
There	O
are	O
three	O
natural	O
ways	O
to	O
achieve	O
this	O
:	O
as	O
a	O
lower-triangular	B-Algorithm
matrix	I-Algorithm
,	O
an	O
upper-triangular	B-Algorithm
matrix	I-Algorithm
,	O
or	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
There	O
are	O
other	O
ways	O
in	O
which	O
Pascal	O
's	O
triangle	O
can	O
be	O
put	O
into	O
matrix	B-Architecture
form	O
,	O
but	O
these	O
are	O
not	O
easily	O
extended	O
to	O
infinity	O
.	O
</s>
<s>
The	O
non-zero	O
elements	O
of	O
a	O
Pascal	B-Algorithm
matrix	I-Algorithm
are	O
given	O
by	O
the	O
binomial	O
coefficients	O
:	O
</s>
<s>
From	O
this	O
it	O
is	O
easily	O
seen	O
that	O
all	O
three	O
matrices	O
have	O
determinant	O
1	O
,	O
as	O
the	O
determinant	O
of	O
a	O
triangular	B-Algorithm
matrix	I-Algorithm
is	O
simply	O
the	O
product	O
of	O
its	O
diagonal	O
elements	O
,	O
which	O
are	O
all	O
1	O
for	O
both	O
Ln	O
and	O
Un	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
matrices	O
Sn	O
,	O
Ln	O
,	O
and	O
Un	O
are	O
unimodular	B-Algorithm
,	O
with	O
Ln	O
and	O
Un	O
having	O
trace	O
n	O
.	O
</s>
<s>
A	O
Pascal	B-Algorithm
matrix	I-Algorithm
can	O
actually	O
be	O
constructed	O
by	O
taking	O
the	O
matrix	B-Architecture
exponential	O
of	O
a	O
special	O
subdiagonal	O
or	O
superdiagonal	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
example	O
below	O
constructs	O
a	O
7×7	O
Pascal	B-Algorithm
matrix	I-Algorithm
,	O
but	O
the	O
method	O
works	O
for	O
any	O
desired	O
n×n	O
Pascal	O
matrices	O
.	O
</s>
<s>
In	O
the	O
construction	O
of	O
symmetric	B-Algorithm
Pascal	O
matrices	O
like	O
that	O
above	O
,	O
the	O
sub	O
-	O
and	O
superdiagonal	O
matrices	O
do	O
not	O
commute	O
,	O
so	O
the	O
(	O
perhaps	O
)	O
tempting	O
simplification	O
involving	O
the	O
addition	O
of	O
the	O
matrices	O
cannot	O
be	O
made	O
.	O
</s>
<s>
A	O
useful	O
property	O
of	O
the	O
sub	O
-	O
and	O
superdiagonal	O
matrices	O
used	O
for	O
the	O
construction	O
is	O
that	O
both	O
are	O
nilpotent	O
;	O
that	O
is	O
,	O
when	O
raised	O
to	O
a	O
sufficiently	O
great	O
integer	O
power	O
,	O
they	O
degenerate	O
into	O
the	O
zero	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
(	O
See	O
shift	B-Algorithm
matrix	I-Algorithm
for	O
further	O
details	O
.	O
)	O
</s>
<s>
As	O
the	O
n×n	O
generalised	O
shift	O
matrices	O
we	O
are	O
using	O
become	O
zero	O
when	O
raised	O
to	O
power	O
n	O
,	O
when	O
calculating	O
the	O
matrix	B-Architecture
exponential	O
we	O
need	O
only	O
consider	O
the	O
first	O
n+1	O
terms	O
of	O
the	O
infinite	O
series	O
to	O
obtain	O
an	O
exact	O
result	O
.	O
</s>
<s>
Interesting	O
variants	O
can	O
be	O
obtained	O
by	O
obvious	O
modification	O
of	O
the	O
matrix-logarithm	O
PL7	O
and	O
then	O
application	O
of	O
the	O
matrix	B-Architecture
exponential	O
.	O
</s>
<s>
The	O
Laguerre-matrix	O
is	O
actually	O
used	O
with	O
some	O
other	O
scaling	O
and/or	O
the	O
scheme	O
of	O
alternating	O
signs	O
.	O
</s>
<s>
The	O
third	O
example	O
below	O
uses	O
the	O
square	O
of	O
the	O
original	O
PL7-matrix	O
,	O
divided	O
by	O
2	O
,	O
in	O
other	O
words	O
:	O
the	O
first-order	O
binomials	O
(binomial(k , 2 )	O
)	O
in	O
the	O
second	O
subdiagonal	O
and	O
constructs	O
a	O
matrix	B-Architecture
,	O
which	O
occurs	O
in	O
context	O
of	O
the	O
derivatives	B-Algorithm
and	O
integrals	O
of	O
the	O
Gaussian	O
error	O
function	O
:	O
</s>
<s>
If	O
this	O
matrix	B-Architecture
is	O
inverted	O
(	O
using	O
,	O
for	O
instance	O
,	O
the	O
negative	O
matrix-logarithm	O
)	O
,	O
then	O
this	O
matrix	B-Architecture
has	O
alternating	O
signs	O
and	O
gives	O
the	O
coefficients	O
of	O
the	O
derivatives	B-Algorithm
(	O
and	O
by	O
extension	O
the	O
integrals	O
)	O
of	O
Gauss	O
 '	O
error-function	O
.	O
</s>
<s>
Another	O
variant	O
can	O
be	O
obtained	O
by	O
extending	O
the	O
original	O
matrix	B-Architecture
to	O
negative	O
values	O
:	O
</s>
