<s>
there	O
exists	O
a	O
cyclic	B-Algorithm
vector	I-Algorithm
in	O
for	O
,	O
meaning	O
that	O
{	O
v	O
,	O
Av	O
,	O
A2v	O
,	O
...	O
,	O
An−1v}	O
is	O
a	O
basis	O
of	O
V	O
.	O
Equivalently	O
,	O
such	O
that	O
V	O
is	O
cyclic	O
as	O
a	O
-module	O
(	O
and	O
)	O
;	O
one	O
says	O
that	O
is	O
non-derogatory	O
.	O
</s>
<s>
Not	O
every	O
square	B-Algorithm
matrix	I-Algorithm
is	O
similar	B-Algorithm
to	O
a	O
companion	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
But	O
every	O
square	B-Algorithm
matrix	I-Algorithm
is	O
similar	B-Algorithm
to	O
a	O
matrix	O
made	O
up	O
of	O
blocks	O
of	O
companion	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
If	O
has	O
distinct	O
roots	O
(	O
the	O
eigenvalues	O
of	O
C(p )	O
)	O
,	O
then	O
C(p )	O
is	O
diagonalizable	B-Algorithm
as	O
follows	O
:	O
</s>
<s>
If	O
has	O
a	O
non-simple	O
root	O
,	O
then	O
C(p )	O
is	O
n't	O
diagonalizable	B-Algorithm
(	O
its	O
Jordan	O
canonical	O
form	O
contains	O
one	O
block	O
for	O
each	O
distinct	O
root	O
)	O
.	O
</s>
<s>
where	O
is	O
the	O
transpose	O
of	O
the	O
companion	B-Algorithm
matrix	I-Algorithm
for	O
the	O
monic	O
polynomial	O
.	O
</s>
<s>
If	O
is	O
invertible	O
then	O
it	O
is	O
possible	O
to	O
decouple	O
it	O
by	O
making	O
a	O
coordinate	O
transformation	O
as	O
described	O
in	O
the	O
section	O
on	O
Diagonalizability	B-Algorithm
.	O
</s>
