Generic Eigenstructures of Hermitian Pencils
2024 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 45, no 1, p. 260-283Article in journal (Refereed) Published
Abstract [en]
We obtain the generic complete eigenstructures of complex Hermitian n x n matrix pencils with rank at most r (with r <= n). To do this, we prove that the set of such pencils is the union of a finite number of bundle closures, where each bundle is the set of complex Hermitian n x n pencils with the same complete eigenstructure (up to the specific values of the distinct finite eigenvalues). We also obtain the explicit number of such bundles and their codimension. The cases r = n, corresponding to general Hermitian pencils, and r < n exhibit surprising differences, since for r < n the generic complete eigenstructures can contain only real eigenvalues, while for r = n they can contain real and nonreal eigenvalues. Moreover, we will see that the sign characteristic of the real eigenvalues plays a relevant role for determining the generic eigenstructures.
Place, publisher, year, edition, pages
Siam Publications , 2024. Vol. 45, no 1, p. 260-283
Keywords [en]
matrix pencil, rank, strict equivalence, congruence, Hermitian matrix pencil, orbit, bundle, closure, sign characteristic
National Category
Mathematics
Identifiers
URN: urn:nbn:se:oru:diva-112807DOI: 10.1137/22M1523297ISI: 001174947800015Scopus ID: 2-s2.0-85186639296OAI: oai:DiVA.org:oru-112807DiVA, id: diva2:1848482
Funder
Swedish Research Council, 2021-05393
Note
he work of the first and third authors was partially supported by the Agencia Estatal de Investigacion of Spain, grants PID2019-106362GB-I00 MCIN/AEI/10.13039/501100011033/and RED2022-134176-T, the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23) , and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation) . The work of the second author was supported by the Swedish Research Council (VR) , grant 2021-05393.
2024-04-032024-04-032024-04-03Bibliographically approved