PreprintarXiv licencearXiv
Low-rank perturbation of symplectic matrices
Abstract
In this paper the generic behavior of the Jordan structure of symplectic matrices under low-rank structure-preserving transformations is considered. So far, only the case of rank-one perturbations was covered in the literature, and this result is now extended to the case of arbitrary rank k. As a key step toward the main result, it is shown that all symplectic matrices that differ from the identity by rank k can be represented as a product of k symplectic reflectors unless they are similar to a diagonalizable matrix having only the eigenvalues ± 1.
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BibTeX
@misc{mehl2026lowrank,
title = {{Low-rank perturbation of symplectic matrices}},
author = {Christian Mehl},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20646},
eprint = {2610.20646},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20646},
}
RIS
TY - UNPB
TI - Low-rank perturbation of symplectic matrices
AU - Christian Mehl
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.20646
UR - https://arxiv.org/abs/2610.20646
AB - In this paper the generic behavior of the Jordan structure of symplectic matrices under low-rank structure-preserving transformations is considered. So far, only the case of rank-one perturbations was covered in the literature, and this result is now extended to the case of arbitrary rank k. As a key step toward the main result, it is shown that all symplectic matrices that differ from the identity by rank k can be represented as a product of k symplectic reflectors unless they are similar to a diagonalizable matrix having only the eigenvalues ± 1.
ER -