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is an ordered basis of (k n )— . Using the fact that [ ——
= ’AT , compute the matrix
A ]E

— — n’1
A (v)| · · · |(
P = [id]Bv ,E — = [v| A) (v)].


Note that if Bv is the dual ordered basis of Bv in k n , then [id]E,Bv = P T by (2.3). Compute


—n —n
( A ) (v), then compute the column vector v = [(
¯ A ) (v)]Bv by solving the matrix equation
—n —
Pv = (
¯ A ) (v). Let Q = [ A ]Bv .
We see that
® 
0 0 ···
0 a0
 
 
 ’1 0 0 · · · 
a1
 
 
Q =  0 ’1 0 · · · ,
a2
 
. 
. . .. .
. 
. . .
.
. . . .
° »
0 0 · · · an’1
0

˜
where v = ’(a0 , . . . , an’1 )T . De¬ne A = ’QT = [
¯ A ]Bv . We claim that Y = AY + B and



˜ ˜ ˜
Y = AY + B are equivalent systems, where B = P T B. Indeed, one checks that the map
ψ : k n+1 ’ k n+1 given by ® 
T
P 0
ψ(Y ) = ° »Y
ˆ ˆ
0 1
is an isomorphism that has the properties required for equivalence of inhomogeneous sys-
˜ ˜
tems. The system Y = AY + B has the form
® 
···
0 1 0 0
 
 
···
0 
0 1 0
Y = .  Y + B,
˜ (3.18)
. 
. . .
..
. 
. . .
.
. . .
° »
’a1 ’a2 ··· ’an’1
’a0

where ai ∈ k. Write Y = (y1 , . . . , yn )T , B = (˜1 , . . . , ˜n )T . This yields
˜ b b

yi = yi+1 + ˜i for 1 ¤ i ¤ n ’ 1
b (3.19)

and
n
aj yj + ˜n .
yn = ’ b (3.20)
j=1

By solving for yi+1 in (3.19) and applying to (3.20), we may eliminate the variable yi+1
from (3.20) for i = n ’ 1, n ’ 2, . . . , 1. If we write y = y1 , we obtain an equation L(y) = b,
n’1
where L = Dn + ai and
i=0
® 
j
n n’1
˜(n’i) + ° ˜(j+1’i) » ∈ k.
b= bi aj bi (3.21)
i=1 j=1 i=1


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˜ ˜ ˜
We claim that Y = AY + B and Y = AY + (0, . . . , 0, b)T are equivalent systems. Indeed,
ˆ
de¬ne a map ψ : k n+1 ’ k n+1 given by

®
w
In
»Y ,
ψ(Y ) = °
ˆˆ ˆ
0 1

where
n
˜(n’i) )T ;
w = (0, ˜1 , ˜1 + ˜2 , ˜1 + ˜2 + ˜3 , . . . ,
bb bb b b bi
i=1

one checks that this mapping is an isomorphism that has the properties required for equiv-
alence of inhomogeneous systems. This shows that Y = AY + B is equivalent to L(y) = b,
as desired.
Given a homogeneous equation L(y) = 0, L = L1 —¦ L2 , L2 = Dm + am’1 Dm’1 + · · · +
˜
a1 D + a0 , L1 = Dn +an’1 Dn’1 + · · · + a1 D +a0 . Consider the following system of equations
˜ ˜
in y1 , . . . , yn+m :
m

ym = ai’1 yi + ym+1
˜
i=1
n

yn+m = ai’1 ym+i
i=1
yi+1 for i ∈ {m, n + m} .
yi = /

Written in matrix form, this system is
® 
AL2 C0
Y =° » Y,
0 AL1

where C0 ∈ k m—n is the matrix having 1 in the m, 1 position and zero everywhere else.
In what follows, we will show how to compute the group of a system of the form
® 
A2 C
Y =° » Y, A2 ∈ k m—m , A1 ∈ k n—n , C ∈ k m—n , (3.22)
0 A1

where Y = Ai Y is completely reducible for i = 1, 2 and C is an arbitrary given matrix.

Proposition 3.3.1 Given a system of the form (3.16). Suppose that the associated homo-
geneous system Y = AY is completely reducible. Let U = Gal(KI /KH ) ⊆ GI . Then the
following statements hold:

1. Fix an arbitary particular solution · ∈ KI of Y = AY + B. Then, the map ¦· : U ’
n


VH given by ¦· („ ) = „.· ’ · is an injective GH -module homomorphism, where the

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action of GH on U (resp., on VH ) is by conjugation (resp., by the usual representation
of Galois group on solution space). In particular, U is a vector group over C.

2. GI is isomorphic to U GH , where GH acts on U by conjugation. This is a Levi
decomposition of GI , i.e., U is the unipotent radical of GI and GI has a maximal
reductive subgroup isomorphic to GH .

Proof. These statements follow from the proof of Proposition 3.2.1, suitably rewritten in
terms of ¬rst-order systems.




The following result is adapted from Lemma 1 and the discussion in Section 2 of [Ber90].
A fundamental solution matrix of a ¬rst-order homogeneous system is a matrix whose column
vectors form a basis of the solution space.

Lemma 3.3.2 Let Y1 (resp., Y2 ) be a fundamental solution matrix of Y = A1 Y (resp.,
Y = A2 Y ). Then, the matrix
« 
U Y1
Y2
Y =  (3.23)
0 Y1

is a fundamental solution matrix of equation (3.22) if and only if U satis¬es

U = A2 U ’ U A1 + C. (3.24)

Proof. This statement is veri¬ed by direct calculation.




The following de¬nitions provide an interpretation of (3.24) in terms of connections. Let
Mi = (D/DLi )— for i = 1, 2. Then [ = Ai , where Ei is a suitable basis of Mi for
Mi ]Ei

i = 1, 2. Let ψ0 : M1 ’ M2 be such that [ψ0 ]E1 ,E2 = C. Applying (3.3), we conclude that
(3.24) is a matrix expression of the equation Hom (φ) = ψ0 , where φ is an unknown member
of Homk (M1 , M2 ).
We compute another matrix expression of this equation as follows: Let K/k be a ¬eld
extension with CK = Ck . Given V ∈ K m—n , let vi be the ith column of V. Then de¬ne
˜ ˜
V ∈ C mn by V = (v1 , . . . , vn )T . Formulas (3.4) and (3.5) imply that V satis¬es V =
T T

˜ ˜ ˜
A2 V ’ V A1 if and only if V satis¬es V = (’AT — Im + In — A2 )V . Furthermore, if Y1
1


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(resp., Y2 ) is a fundamental solution matrix of the system Y = A1 Y (resp., Y = A2 Y ),
then a calculation shows that (Y1’1 )T — Y2 is a fundamental solution matrix of the system
˜ ˜ ˜
V = (’AT — Im + In — A2 )V , where V ∈ C mn is a column vector of unknowns. De¬ne
1

ci ∈ C m to be the ith column vector of C. We see that the equation Hom (φ) = ψ0 also has
the matrix expression

˜ ˜ ˜
V = (’AT — Im + In — A2 )V + C, (3.25)
1


˜
where C = (cT , . . . , cT )T .
1 n


Lemma 3.3.3 Assume that C k. Let Y = A1 Y and Y = A2 Y be completely reducible
systems and let K/k be the Picard-Vessiot extension of k corresponding to equation (3.22).
Let F/k, F ‚ K, be the Picard-Vessiot extension corresponding to
® 
A2 0
Y =° » Y. (3.26)
0 A1

Then, the following statements hold:

1. The system

˜ ˜
V = (’AT — Im + In — A2 )V (3.27)
1


admits a full set of F -rational solutions, so there exists a tower of ¬elds k ⊆ E ⊆ F
with E/k a Picard-Vessiot extension for (3.27). In particular, Gal(E/k) is a quotient
of Gal(F/k).

2. K = F (˜), where · is a particular solution of (3.25).
· ˜

3. E(˜)/k is a Picard-Vessiot extension with Gal(E(˜)/k)
· · W Gal(E/k), where W is a

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