Page 137 - Textos de Matemática Vol. 39

MATRIX PENCILS 127
be a nonsingular matrix. For each matrix pencil Cx+D, where C,D ∈ Fm×n, let
PS(Cx + D) = (s1,1C + s2,1D)x + (s1,2C + s2,2D). For every polynomial h(x, y) ∈ F [x, y], let
ΠS (h) = h(s1,1x + s1,2y, s2,1x + s2,2y).
The operators PS and ΠS where introduced in [2] and they have some nice
properties:
• ForanypencilCx+D,PS−1(PS(Cx+D))=Cx+D.
• Foranypolynomialh(x,y)∈F[x,y],ΠS−1(ΠS(h))=h.
• If h(x,y),h′(x,y) ∈ F[x,y], then h | h′ if and only if ΠS(h) | ΠS(h′).
• If δ1 | · · · | δr are the homogeneous invariant factors of a pencil Cx+D,
then ΠS(δ1) | ··· | ΠS(δr) are the homogeneous invariant factors of
PS(Cx+D).
• If Cx+D and C′x+D′ are strictly equivalent pencils, then PS(Cx+D)
and PS(C′x + D′) are strictly equivalent. With every monic polynomial
f(x)=xk +ak−1xk−1 +···+a0 ∈F[x], we associate the homogeneous polynomial
˜
f(x,y)=xk +ak−1xk−1y+···+a0yk ∈F[x,y].
This operation has the following properties:
• For every monic polynomials f,g ∈ F[x], f | g if and only if f˜| g˜.
• Let Cx + D be a matrix pencil without infinite elementary divisors. Then β1 | ··· | βr are the invariant factors of Cx + D if and only if
β˜1 | · · · | β˜r are the homogeneous invariant factors of Cx + D.
Using the operators above, it can be deduced, as a corollary to Theo- rem 1.2, that the homogeneous invariant factors of a pencil and the homoge- neous invariant factors of a subpencil interlace.
Theorem 1.3. Let P (x) ∈ F [x]m×n and Q(x) ∈ F [x]p×q be matrix pencils, with p ≤ m and q ≤ n. Let δ1 | ··· | δr be the homogeneous invariant factors of P (x) and let γ1 | · · · | γs be the homogeneous invariant factors of Q(x).
If there exists a matrix pencil strictly equivalent to P(x) containing Q(x) as a subpencil, then s ≤ r ≤ s + m + n − p − q,
(1.5) δi | γi, for every i ∈ {1,...,s}, and
(1.6) γi | δi+m+n−p−q,
for every i ∈ {1, . . . , s} such that i + m + n − p − q ≤ r.