Page 136 - Textos de Matemática Vol. 39

126 ALICIA ROCA AND FERNANDO C. SILVA
classes of a square matrix with a prescribed submatrix and has inspired many results of the same type. The possible similarity classes of a square matrix with prescribed rows were described in [17]. The possible similarity classes of a square matrix with a prescribed nonprincipal submatrix were described in [7, 14, 18]. Using previous results, the possible similarity classes of a square matrix with a prescribed submatrix in an arbitrary position were described in [3]. However, this last result was only proved for matrices over infinite fields and the necessary and sufficient conditions involve an existential quantifier that makes it less elegant. A new proof, valid over arbitrary fields, was given in [11, 12], but to find new conditions without the existencial quantifier remains an open problem.
The divisibility conditions (1.1)–(1.2) became known as the “interlacing inequalities for invariant factors”. In general, the invariant factors of a polyno- mial matrix P (x) and the invariant factors of a submatrix of P (x) interlace, as described below.
Theorem 1.2. [13, 15] Let P(x) ∈ F[x]m×n, Q(x) ∈ F[x]p×q, with p ≤ m and q≤n.Letβ1 |···|βr betheinvariantfactorsofP(x)andletα1 |···|αs be the invariant factors of Q(x).
There exists a matrix equivalent to P(x) containing Q(x) as a submatrix if and only if s ≤ r ≤ s + m + n − p − q,
(1.3) βi | αi, for every i ∈ {1,...,s}, and
(1.4) αi | βi+m+n−p−q,
for every i ∈ {1, . . . , s} such that i + m + n − p − q ≤ r.
TwomatrixpencilsA1x+B1 andA2x+B2,whereA1,B1,A2,B2 ∈Fm×n, are said to be strictly equivalent if there exist nonsingular matrices U ∈ F m×m and V ∈ Fn×n such that A2x+B2 = U(A1x+B1)V. As C1,C2 ∈ Fm×m are similar if and only if Im x − C1 and Im x − C2 are strictly equivalent, strict equivalence of matrix pencils may be viewed as a generalization of similarity of constant matrices. A canonical form for strict equivalence was obtained by Kronecker by the end of the nineteenth century. With a pencil Ax + B, where A, B ∈ F m×n, we associate a homogeneous pencil Ax + By, with entries in the ring of polynomials in two distinct indeterminates x,y. The invariant factors of a homogeneous pencil Ax + By are homogeneous polynomials and are called the homogeneous invariant factors of Ax + B. See [6] for details.
Let
S=s1,1 s1,2 ∈F2×2 s2,1 s2,2