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Bundles of matrix pencils (under strict equivalence) are sets of pencilshaving the same Kronecker canonical form, up to the eigenvalues (namely, theyare an infinite union of orbits under strict equivalence). The notion of bundlefor matrix pencils was introduced in the 1990's, following the same notion formatrices under similarity, introduced by Arnold in 1971, and it has beenextensively used since then. Despite the amount of literature devoted todescribing the topology of bundles of matrix pencils, some relevant questionsremain still open in this context. For example, the following two: (a) providea characterization for the inclusion relation between the closures (in thestandard topology) of bundles; and (b) are the bundles open in their closure?The main goal of this paper is providing an explicit answer to these twoquestions. In order to get this answer, we also review and/or formalize somenotions and results already existing in the literature. We also prove thatbundles of matrices under similarity, as well as bundles of matrix polynomials(defined as the set of $m\times n$ matrix polynomials of the same grade havingthe same spectral information, up to the eigenvalues) are open in theirclosure.

On bundles of matrix pencils under strict equivalence