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Low rank perturbation of Weierstrass structure. (English) Zbl 1163.15010

Some assertions of the following types are proved: Let \(A_0 + \lambda A_1\) be a regular matrix pencil, and let \(\lambda_0\) be one of its finite eigenvalues having \(g\) elementary Jordan blocks in the Weierstrass canonical form. Then it is shown that for the most matrices \(B_0\) and \(B_1\) with \(\text{rank}(B_0+\lambda_0 B_1)< g\) there are \(g -\text{rank}(B_0+ \lambda_0 B_1)\) Jordan blocks corresponding to the eigenvalue \(\lambda_0\) in the Weierstrass form of the perturbed pencil \(A_0 + B_0 +\lambda (A_1 + B_1)\). If \(A_0 + \lambda A_1\) has an infinite eigenvalue, then the corresponding result follows from considering the zero eigenvalue of the dual pencils \(A_1 + \lambda A_0\) and \(A_1 + B_1 + \lambda (A_0 + B_0)\).

MSC:

15A22 Matrix pencils
15A18 Eigenvalues, singular values, and eigenvectors
15A21 Canonical forms, reductions, classification
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