<s>
More	O
generally	O
,	O
a	O
nilpotent	B-Algorithm
transformation	I-Algorithm
is	O
a	O
linear	B-Architecture
transformation	I-Architecture
of	O
a	O
vector	O
space	O
such	O
that	O
for	O
some	O
positive	O
integer	O
(	O
and	O
thus	O
,	O
for	O
all	O
)	O
.	O
</s>
<s>
More	O
generally	O
,	O
any	O
-dimensional	O
triangular	B-Algorithm
matrix	I-Algorithm
with	O
zeros	O
along	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
is	O
nilpotent	O
,	O
with	O
index	O
.	O
</s>
<s>
Although	O
the	O
examples	O
above	O
have	O
a	O
large	O
number	O
of	O
zero	O
entries	O
,	O
a	O
typical	O
nilpotent	B-Algorithm
matrix	I-Algorithm
does	O
not	O
.	O
</s>
<s>
Perhaps	O
some	O
of	O
the	O
most	O
striking	O
examples	O
of	O
nilpotent	B-Algorithm
matrices	I-Algorithm
are	O
square	B-Algorithm
matrices	I-Algorithm
of	O
the	O
form	O
:	O
</s>
<s>
The	O
derivative	B-Algorithm
operator	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
.	O
</s>
<s>
We	O
know	O
that	O
applying	O
the	O
derivative	B-Algorithm
to	O
a	O
polynomial	O
decreases	O
its	O
degree	O
by	O
one	O
,	O
so	O
when	O
applying	O
it	O
iteratively	O
,	O
we	O
will	O
eventually	O
obtain	O
zero	O
.	O
</s>
<s>
Therefore	O
,	O
on	O
such	O
a	O
space	O
,	O
the	O
derivative	B-Algorithm
is	O
representable	O
by	O
a	O
nilpotent	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
For	O
an	O
square	B-Algorithm
matrix	I-Algorithm
with	O
real	O
(	O
or	O
complex	O
)	O
entries	O
,	O
the	O
following	O
are	O
equivalent	O
:	O
</s>
<s>
The	O
index	O
of	O
an	O
nilpotent	B-Algorithm
matrix	I-Algorithm
is	O
always	O
less	O
than	O
or	O
equal	O
to	O
.	O
</s>
<s>
For	O
example	O
,	O
every	O
nilpotent	B-Algorithm
matrix	I-Algorithm
squares	O
to	O
zero	O
.	O
</s>
<s>
The	O
determinant	O
and	O
trace	O
of	O
a	O
nilpotent	B-Algorithm
matrix	I-Algorithm
are	O
always	O
zero	O
.	O
</s>
<s>
Consequently	O
,	O
a	O
nilpotent	B-Algorithm
matrix	I-Algorithm
cannot	O
be	O
invertible	O
.	O
</s>
<s>
The	O
only	O
nilpotent	O
diagonalizable	B-Algorithm
matrix	I-Algorithm
is	O
the	O
zero	O
matrix	O
.	O
</s>
<s>
Consider	O
the	O
(	O
upper	O
)	O
shift	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
As	O
a	O
linear	B-Architecture
transformation	I-Architecture
,	O
the	O
shift	B-Algorithm
matrix	I-Algorithm
"	O
shifts	O
"	O
the	O
components	O
of	O
a	O
vector	O
one	O
position	O
to	O
the	O
left	O
,	O
with	O
a	O
zero	O
appearing	O
in	O
the	O
last	O
position	O
:	O
</s>
<s>
This	O
matrix	O
is	O
nilpotent	O
with	O
degree	O
,	O
and	O
is	O
the	O
canonical	O
nilpotent	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
where	O
each	O
of	O
the	O
blocks	O
is	O
a	O
shift	B-Algorithm
matrix	I-Algorithm
(	O
possibly	O
of	O
different	O
sizes	O
)	O
.	O
</s>
<s>
That	O
is	O
,	O
if	O
is	O
any	O
nonzero	O
22	O
nilpotent	B-Algorithm
matrix	I-Algorithm
,	O
then	O
there	O
exists	O
a	O
basis	O
b1	O
,	O
b2	O
such	O
that	O
Nb1	O
=	O
0	O
and	O
Nb2	O
=	O
b1	O
.	O
</s>
<s>
The	O
signature	O
characterizes	O
up	O
to	O
an	O
invertible	O
linear	B-Architecture
transformation	I-Architecture
.	O
</s>
<s>
Conversely	O
,	O
any	O
sequence	O
of	O
natural	O
numbers	O
satisfying	O
these	O
inequalities	O
is	O
the	O
signature	O
of	O
a	O
nilpotent	B-Algorithm
transformation	I-Algorithm
.	O
</s>
