Matrix pencils completion problems. II
[For part I see the author, ibid. 428, No.~1, 259--304 (2008; Zbl 1131.15010).] The paper deals with the matrix pencils completion problem. This problem analyzes the existence of a matrix pencil with prescribed Kronecker invariants and a regular subpencil. Specifically, the author studies the following problem: Let \(A(\lambda) \in F[\lambda]^{n \times n}\) be a regular matrix pencil. Let \(E(\lambda) \in F[\lambda]^{(n+x) \times (n+y)}\) be a matrix pencil. Find necessary and sufficient conditions for the existence of pencils \(B(\lambda) \in F[\lambda]^{n \times x}\), \(C(\lambda) \in F[\lambda]^{y \times n}\) and \(D(\lambda) \in F[\lambda]^{y \times x}\) such that the pencil \[ \left( \begin{matrix} A(\lambda) & B(\lambda) \\ C(\lambda) & D(\lambda) \end{matrix} \right) \] is strictly equivalent to \(E(\lambda)\). The author explicitly solves the above problem, by giving necessary and sufficient conditions without any existential quantifiers involved. On the other hand, in the second part of this paper the following problem is analyzed: Let \(A(\lambda) \in F[\lambda]^{(n+p) \times (n+m)}\) and \(E(\lambda) \in F[\lambda]^{(n+p+m+x) \times (n+m+p+y)}\) be matrix pencils. Find necessary and sufficient conditions for the existence of pencils \(B(\lambda) \in F[\lambda]^{(n+p) \times (p+x)}\), \(C(\lambda) \in F[\lambda]^{(m+y) \times (n+m)}\) and \(D(\lambda) \in F[\lambda]^{(m+y) \times (x+p)}\) such that the pencil \[ M(\lambda)=\left( \begin{matrix} A(\lambda) & B(\lambda) \\ C(\lambda) & D(\lambda) \end{matrix}\right) \] is strictly equivalent to \(E(\lambda)\) and such that \(\text{rank}[A(\lambda) \;\;B(\lambda)]=n+p\) and \(\text{rank}[A(\lambda) \;\;C(\lambda)]^T=n+m\). The author gives a complete solution of the above problem and she proves that this problem is equivalent to the problem of describing the possible Kronecker invariants of a pencil \(M(\lambda)\) with a prescribed arbitrary subpencil \(A(\lambda)\), if there exists a regular intermediate pencil, i.e. if there exists a regular \((n+m+p)\times (n+m+p)\) subpencil of \(M(\lambda)\) containing \(A(\lambda)\) as a subpencil.
scientific article; zbMATH DE number 5289140