Page 135 - Textos de Matemática Vol. 39

MATRIX PENCILS WITH A PRESCRIBED SUBPENCIL
ALICIA ROCA AND FERNANDO C. SILVA
Dedicated to Eduardo Marques de Sa´ on the occasion of his 60th birthday.
Abstract. A theorem due to S´a and Thompson, published more than a quar- ter of a century ago, described the possible similarity classes of a square matrix, when a principal submatrix is prescribed. This theorem has inspired other re- sults on matrix completions. In this paper, we mention some of these other results and we study the possible strict equivalence classes of a pencil with a prescribed zero subpencil.
1. Introduction
Let F be a field.
A theorem due to S´a [13] and Thompson [15], published more than a quarter of a century ago, described the possible similarity classes of a square matrix B ∈ Fm×m, when a principal submatrix is prescribed. Recall that the similarity class of a square matrix B is completely characterized by the invariant factors of the matrix pencil Imx − B, that has entries in the ring F [x] of the polynomials in the indeterminate x with coefficients in F . For notational convenience, we shall assume that the invariant factors of a matrix over F[x] are always monic.
Theorem 1.1. [13, 15] Let B∈Fm×m, A∈Fp×p, with p ≤ m. Let β1 | ··· | βm be the invariant factors of Imx − B and let α1 | · · · | αp be the invariant factors of Ipx − A.
There exists a matrix similar to B containing A as a principal submatrix if and only if
(1.1) βi | αi, for every i ∈ {1,...,p}, and
(1.2) αi | βi+2(m−p),
for every i∈{1,...,p} such that i+2(m−p)≤m.
Some previous results had already described the possible eigenvalues of a square matrix with a prescribed submatrix, e. g., [9, 10, 16]. But, as far as we know, Theorem 1.1 was the first one to describe the possible similarity
2000 Mathematics Subject Classification. 15A22
Key words and phrases. Matrix pencils, Completion problems, Invariant factors.
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