<s>
In	O
mathematics	O
,	O
a	O
shift	B-Algorithm
matrix	I-Algorithm
is	O
a	O
binary	B-Algorithm
matrix	I-Algorithm
with	O
ones	O
only	O
on	O
the	O
superdiagonal	O
or	O
subdiagonal	O
,	O
and	O
zeroes	O
elsewhere	O
.	O
</s>
<s>
A	O
shift	B-Algorithm
matrix	I-Algorithm
U	O
with	O
ones	O
on	O
the	O
superdiagonal	O
is	O
an	O
upper	O
shift	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
alternative	O
subdiagonal	O
matrix	O
L	O
is	O
unsurprisingly	O
known	O
as	O
a	O
lower	O
shift	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Clearly	O
,	O
the	O
transpose	O
of	O
a	O
lower	O
shift	B-Algorithm
matrix	I-Algorithm
is	O
an	O
upper	O
shift	B-Algorithm
matrix	I-Algorithm
and	O
vice	O
versa	O
.	O
</s>
<s>
As	O
a	O
linear	O
transformation	O
,	O
a	O
lower	O
shift	B-Algorithm
matrix	I-Algorithm
shifts	O
the	O
components	O
of	O
a	O
column	O
vector	O
one	O
position	O
down	O
,	O
with	O
a	O
zero	O
appearing	O
in	O
the	O
first	O
position	O
.	O
</s>
<s>
An	O
upper	O
shift	B-Algorithm
matrix	I-Algorithm
shifts	O
the	O
components	O
of	O
a	O
column	O
vector	O
one	O
position	O
up	O
,	O
with	O
a	O
zero	O
appearing	O
in	O
the	O
last	O
position	O
.	O
</s>
<s>
Premultiplying	O
a	O
matrix	O
A	O
by	O
a	O
lower	O
shift	B-Algorithm
matrix	I-Algorithm
results	O
in	O
the	O
elements	O
of	O
A	O
being	O
shifted	O
downward	O
by	O
one	O
position	O
,	O
with	O
zeroes	O
appearing	O
in	O
the	O
top	O
row	O
.	O
</s>
<s>
Postmultiplication	O
by	O
a	O
lower	O
shift	B-Algorithm
matrix	I-Algorithm
results	O
in	O
a	O
shift	O
left	O
.	O
</s>
<s>
Similar	B-Algorithm
operations	O
involving	O
an	O
upper	O
shift	B-Algorithm
matrix	I-Algorithm
result	O
in	O
the	O
opposite	O
shift	O
.	O
</s>
<s>
Clearly	O
all	O
finite-dimensional	O
shift	O
matrices	O
are	O
nilpotent	O
;	O
an	O
n	O
by	O
n	O
shift	B-Algorithm
matrix	I-Algorithm
S	O
becomes	O
the	O
null	B-Algorithm
matrix	I-Algorithm
when	O
raised	O
to	O
the	O
power	O
of	O
its	O
dimension	O
n	O
.	O
</s>
<s>
where	O
each	O
of	O
the	O
blocks	O
S1	O
,	O
S2	O
,...,	O
Sr	O
is	O
a	O
shift	B-Algorithm
matrix	I-Algorithm
(	O
possibly	O
of	O
different	O
sizes	O
)	O
.	O
</s>
<s>
Clearly	O
there	O
are	O
many	O
possible	O
permutations	B-Algorithm
.	O
</s>
