Page 140 - Textos de Matemática Vol. 39

130 ALICIA ROCA AND FERNANDO C. SILVA
when the prescribed subpencil corresponds to complete rows of the bigger pen- cil. Several earlier results about matrix completions, known from the literature, are then recovered from this theorem.
Recently, we have started studying the possible strict equivalence classes of a pencil with a prescribed constant subpencil and are still working on this question. In this paper, we include a partial result.
2. Main Result
Let A,B ∈ Fm×n. Let p ∈ {1,...,m}, q ∈ {1,...,n}. Throughout this
section, we shall use the following notation:
• t is the number of infinite elementary divisors of Ax + B.
• u (respectively, v) is the number of nonzero column (respectively, row)
minimal indices of Ax + B.
• jc (respectively, jr ) is the largest nonnegative integer j such that Ax+B
has j nonzero column (respectively, row) minimal indices whose sum does not exceed m − p (respectively, n − q).
We shall prove the following theorem.
Theorem 2.1. Suppose that F is algebraically closed. There exists a matrix pencil strictly equivalent to Ax + B containing a zero subpencil of size p × q if and only if
(2.1) rank(Ax+B)≤m+n−p−q+jc −u, (2.2) rank(Ax+B)≤m+n−p−q+jr −v.
Given a matrix pencil H, denote by dc(H) (respectively, dr(H)) the di- mension of the F-space generated by the columns (respectively, rows) of H. Clearly, if H and H′ are strictly equivalent pencils, then dc(H) = dc(H′) and dr(H) = dr(H′). Bearing in mind the Kronecker canonical form, it is easy to see that
(2.3) dc(Ax + B) = rank(Ax + B) + u, dr(Ax + B) = rank(Ax + B) + v.
Lemma 2.2. Let H = [ G ∗ ] ∈ F[x]m×h be a matrix pencil, where G∈F[x]m×g.Letk1 ≤···≤ks andc1 ≤···≤ct bethecolumnminimal indices of H and G, respectively. Then t ≤ s and ki ≤ ci, i ∈ {1,...,t}.
Proof. Ass=h−rankH andt=g−rankG,weconcludethatt≤s.
Let X1,. . . ,Xs and Y1,. . . ,Yt be fundamental series of solutions of HX = 0 and GY = 0, respectively (see [6] for a definition). For every i ∈ {1, . . . , t}, let
Zi = Yi ∈F[x]h×1. 0