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.

我们详细说明了两个线性关系的约旦结构的偏差，它们是彼此的有限维扰动。我们比较它们的长度至少为n的约旦链的数量。在算子的情况下，最近证明这些数的差与n无关，并且最多是算子之间的缺陷。本文的主要结果之一表明，在线性关系的情况下，该数字必须乘以\（n + 1 \）而且这个界限很尖锐。这种行为的原因是存在奇异链。我们将我们的结果应用于奇异和规则矩阵铅笔的一维扰动。这是通过线性关系表示矩阵铅笔来完成的。这种技术既可以证明普通铅笔的已知结果，也可以证明奇异铅笔的新结果。