Abstract
We elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare their number of Jordan chains of length at least n. In the operator case, it was recently proved that the difference of these numbers is independent of n and is at most the defect between the operators. One of the main results of this paper shows that in the case of linear relations this number has to be multiplied by n+1 and that this bound is sharp. The reason for this behavior is the existence of singular chains. We apply our results to one-dimensional perturbations of singular and regular matrix pencils. This is done by representing matrix pencils via linear relations. This technique allows for both proving known results for regular pencils as well as new results for singular ones.
Highlights
Given a pair of matrices E, F ∈ Cd×d, the associated matrix pencil is defined by P(s) := s E − F. (1.1)The theory of matrix pencils occupies an increasingly important place in linear algebra, due to its numerous applications
The matrix pencil P is called regular if det(s E − F) is not identically zero, and it is called singular otherwise
Perturbation theory for regular matrix pencils P(s) := s E − F is a well developed field, we mention here only [14,21,36,45] which is a short list of papers devoted to this subject
Summary
Given a pair of matrices E, F ∈ Cd×d , the associated matrix pencil is defined by. P(s) := s E − F. In this form it can be found in [21], but it is mainly due to [14,45] The proof of this inequality, as many other results concerning perturbation theory for regular matrix pencils, is based on a detailed analysis of the determinant. There, the generic change in the Kronecker canonical form of a singular pencil under low-rank perturbations resulting again in a singular pencil is considered. We develop a different approach to treat finite rank perturbations of singular matrix pencils. This is done by representing matrix pencils via linear relations, see [6,7,11]. P and P + Q are rankone perturbations of each other, which means that they differ by a rank-one matrix
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