where V is a finite dimensional vector space over F. Define the
multiplication of something in D with something in V as follows.
g(x)v ≡ g(L)v, L0 ≡ I
As mentioned above, V is a finitely generated torsion module and F
[x ]
is a p.i.d. The non-invertible primes
are polynomials irreducible over F. Letting
{z1,⋅⋅⋅,zl}
be a basis for V, it follows that
V = Dz1 + ⋅⋅⋅+ Dzl
and so there exist irreducible polynomials over F,
{ϕi(x)}
_{i=1}^{n} and corresponding positive integers k_{i} such
that
( k1) ( kn)
V = ker ϕ1 (L) ⊕ ⋅⋅⋅⊕ ker ϕn (L )
This is just Theorem 6.1.10 obtained as a special case. Recall that the entire theory of canonical forms is
based on this result. This follows because
{ }
Mϕi = v ∈ V : ϕi(x)ki v ≡ ϕi(L)ki v = 0 .
Now continue letting D = F
[x]
,L ∈ℒ
(V, V)
for V a finite dimensional vector space and
g
(x)
v ≡ g
(L)
v as above. Consider M_{ϕi}≡ ker
( ki)
ϕi(L )
which is a sub module of V . Then by Theorem
9.5.9, this is the direct sum of cyclic sub modules.
M ϕi = Dv1 ⊕ ⋅⋅⋅⊕ Dvs (*)
(*)
where s = rank
(M )
ϕi
.
At this point, note that ann
(M ϕi)
=
( ki)
ϕi(x)
and so ann
(Dvj)
=
( lj)
ϕi(x)
where l_{j}≤ k_{i}. If d_{i} is
the degree of ϕ_{i}
(x )
, this implies that for v_{j} being one of the v in ∗,
2 l d−1
1,Lvj,L vj,⋅⋅⋅,L ji vj
must be a linearly independent set since otherwise, ann
(Dvj)
≠
( )
ϕi(x)lj
because a polynomial of smaller
degree than the degree of ϕ_{i}
(x)
^{lj} will annihilate v_{j}. Also,
( )
Dvj ⊆ span vj,Lvj,L2vj,⋅⋅⋅,Lljdi− 1vj
by division. Vectors in Dv_{j} are of the form g
(x )
v_{j} =
( )
ϕ (x)lj k(x)+ ρ(x)
i
v_{j} = ρ
(x)
v_{j} where the degree
of ρ
(x )
is less than l_{j}d_{i}. Thus a basis for Dv_{j} is
{ 2 ljdi−1 }
vj,Lvj,L vj,⋅⋅⋅,L vj
and the span of these
vectors equals Dv_{j}. Now you see why the term “cyclic” is appropriate for the submodule Dv. This shows
the following theorem.
Theorem 9.7.1Let V be a finite dimensionalvector space over a field of scalars F. Also suppose theminimum polynomial is∏_{i=1}^{n}
(ϕi(x))
^{ki}where k_{i}is a positive integer and the degree of ϕ_{i}
(x)
is d_{i}. Then
( ) ( )
V = ker ϕ1(L)k1 ⊕ ⋅⋅⋅⊕ ker ϕn(L)kn
≡ M ϕ1(x) ⊕ ⋅⋅⋅⊕M ϕn(x)
Furthermore, for each i, in ker
( ki)
ϕi(L )
, there are vectors v_{1},
⋅⋅⋅
,v_{si}and positive integers l_{1},
⋅⋅⋅
,l_{si}eachno larger than k_{i}such that a basis for ker
( )
ϕ (L)ki
i
is given by
{ }
βl1vdi−1,⋅⋅⋅,βlsvisdi−1
1 i
where the symbol β_{vj}^{ljdi−1}signifies the ordered basis
( 2 ljdi−2 ljdi−1 )
vj,Lvj,L vj,⋅⋅⋅,L vj,L vj
Its length is the degree of ϕ_{j}
(x)
^{kj}and is therefore, determined completely by the l_{j}. Thus the lengths ofthe β_{vj}^{ljdi−1}are uniquely determined if they are listed in order of increasing or decreasinglength.
The last claim of this theorem will mean that the various canonical forms are uniquely
determined.
It is clear that the span of β_{vj}^{ljdi−1} is invariant with respect to L because, as discussed
above, this span is Dv_{j} where D = F
[x]
. Also recall that ann
(Dvj )
=
( l)
ϕi(x)j
where l_{j}≤ k_{i}.
Let
l
ϕi(x)j = xljdi + an− 1xljdi−1 + ⋅⋅⋅+ a1x+ a0
Recall that the minimum polynomial has leading coefficient equal to 1. Of course this makes no difference
in the above presentation because a_{n} is invertible and so the ideals and above direct sum are the same
regardless of whether this leading coefficient equals 1, but it is convenient to let this happen since
otherwise, the blocks for the rational canonical form will not be standard. Then what is the matrix of L
restricted to Dv_{j}?
( ljdi−1 ljdi ) ( ljdi−2 ljdi− 1 )
Lvj ⋅⋅⋅ L vj L vj = vj ⋅⋅⋅ L vj L vj M
It follows that the matrix of L with respect to the basis obtained as above will be a block diagonal with
blocks like the above. This is the rational canonical form.
Of course, those blocks corresponding to ker
( ϕ (L )ki)
i
can be arranged in any order by just listing the
β_{v1}^{l1di−1},
⋅⋅⋅
,β_{v
si}^{lsidi−1} in various orders. If we want the blocks to be larger in the top left and get smaller
towards the lower right, we just re-number it to have l_{i} be a decreasing sequence. Note that
this is the same as saying that ann
(Dv1 )
⊆ann
(Dv2)
⊆
⋅⋅⋅
⊆ann
(Dvsi)
. If we want the
blocks to be increasing in size from the upper left corner to the lower right, this corresponds to
re-numbering such that ann
(Dv1)
⊇ann
(Dv2 )
⊇
⋅⋅⋅
⊇ann
(Dv1)
. This second one involves letting
l_{1}≤ l_{2}≤
⋅⋅⋅
≤ l_{si}.
What about uniqueness of the rational canonical form given an order of the spaces ker
( )
ϕi(L)ki
and
under the convention that the blocks associated with ker
( ki)
ϕi(L)
should be increasing or degreasing in
size from upper left toward lower right? In other words, suppose you have
( ki)
ker ϕi(L) = Dv1 ⊕ ⋅⋅⋅⊕ Dvs = Dw1 ⊕ ⋅⋅⋅⊕ Dwt
such that
ann (Dv1) ⊆ ann (Dv2) ⊆ ⋅⋅⋅ ⊆ ann (Dvs )
and
ann(Dw1 ) ⊆ ann (Dw2) ⊆ ⋅⋅⋅ ⊆ ann (Dwt)
will it happen that s = t and the that the blocks associated with corresponding Dv_{i} and Dw_{i} are the same
size? In other words, if ann
(Dvj )
=
( )
ϕi (x)lj
and ann
(Dwj)
=
(ϕi(x )mj)
is m_{j} = l_{j}. If this is so, then
this proves uniqueness of the rational canonical form up to order of the blocks. However, this was proved
above in the discussion on uniqueness, Theorem 9.6.1.
In the case that the minimum polynomial splits the following is also obtained.
Corollary 9.7.2Let V be a finite dimensional vector space over a field of scalars F. Also let the minimalpolynomial be∏_{i=1}^{n}
(x− μi)
^{ki}where k_{i}is a positive integer. Then
( k1) ( kn)
V = ker (L − μ1I) ⊕ ⋅⋅⋅⊕ ker (L − μnI)
Furthermore, for each i, in ker
( )
(L − μiI)ki
, there are vectors v_{1},
⋅⋅⋅
,v_{si}and positive integers l_{1},
⋅⋅⋅
,l_{si}each no larger than k_{i}such that a basis for ker
( ki)
(L − μiI)
is given by
{ }
βlv11−1,⋅⋅⋅,βlsvisi−1
where the symbol β_{vj}^{lj−1}signifies the ordered basis
((L − μ I)lj−1v ,(L − μ I)lj−2v ,⋅⋅⋅,(L − μ I)2v ,(L − μ )v ,v)
i j i j i j i j j
(Note how this is the reverse order to the above. This is to follow the usual convention in the Jordan formin which the string of ones is on the super diagonal.)
. Thus, with respect to this basis, the block associated
with L and β_{vj}^{lj−1} is just
( )
| μi 1 0 |
|| μi ... ||
|| .. ||
( . 1 )
0 μi
This has proved the existence of the Jordan form. You have a string of blocks like the above for
ker
(L − μiI)
^{ki}. Of course, these can be arranged so that the size of the blocks is decreasing from
upper left to lower right. As with the rational canonical form, once it is decided to have the
blocks be decreasing (increasing) in size from upper left to lower right, the Jordan form is
unique.