Spectra inhabiting the left half-plane that are universally realizable
Let Λ = {λ1, λ2, . . ., λn} be a list of complex numbers. Λ is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. Minc ([21],1981) showed that if Λ is diagonalizably pos...
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| Format: | Article |
| Language: | English |
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De Gruyter
2021-12-01
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| Series: | Special Matrices |
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| Online Access: | https://doi.org/10.1515/spma-2021-0155 |
| _version_ | 1818018023429111808 |
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| author | Soto Ricardo L. |
| author_facet | Soto Ricardo L. |
| author_sort | Soto Ricardo L. |
| collection | DOAJ |
| description | Let Λ = {λ1, λ2, . . ., λn} be a list of complex numbers. Λ is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. Minc ([21],1981) showed that if Λ is diagonalizably positively realizable, then Λ is universally realizable. The positivity condition is essential for the proof of Minc, and the question whether the result holds for nonnegative realizations has been open for almost forty years. Recently, two extensions of the Minc’s result have been proved in ([5], 2018) and ([12], 2020). In this work we characterize new left half-plane lists (λ1 > 0, Re λi ≤ 0, i = 2, . . ., n) no positively realizable, which are universally realizable. We also show new criteria which allow to decide about the universal realizability of more general lists, extending in this way some previous results. |
| first_indexed | 2024-04-14T07:34:45Z |
| format | Article |
| id | doaj.art-54f92bab346343a8afe21860ec5d0c1f |
| institution | Directory Open Access Journal |
| issn | 2300-7451 |
| language | English |
| last_indexed | 2024-04-14T07:34:45Z |
| publishDate | 2021-12-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Special Matrices |
| spelling | doaj.art-54f92bab346343a8afe21860ec5d0c1f2022-12-22T02:05:43ZengDe GruyterSpecial Matrices2300-74512021-12-0110118019210.1515/spma-2021-0155Spectra inhabiting the left half-plane that are universally realizableSoto Ricardo L.0Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Chile, Casilla 1280, Antofagasta, ChileLet Λ = {λ1, λ2, . . ., λn} be a list of complex numbers. Λ is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. Minc ([21],1981) showed that if Λ is diagonalizably positively realizable, then Λ is universally realizable. The positivity condition is essential for the proof of Minc, and the question whether the result holds for nonnegative realizations has been open for almost forty years. Recently, two extensions of the Minc’s result have been proved in ([5], 2018) and ([12], 2020). In this work we characterize new left half-plane lists (λ1 > 0, Re λi ≤ 0, i = 2, . . ., n) no positively realizable, which are universally realizable. We also show new criteria which allow to decide about the universal realizability of more general lists, extending in this way some previous results.https://doi.org/10.1515/spma-2021-0155nonnegative matrixuniversal realizabilityjordan structure15a1815a29 |
| spellingShingle | Soto Ricardo L. Spectra inhabiting the left half-plane that are universally realizable Special Matrices nonnegative matrix universal realizability jordan structure 15a18 15a29 |
| title | Spectra inhabiting the left half-plane that are universally realizable |
| title_full | Spectra inhabiting the left half-plane that are universally realizable |
| title_fullStr | Spectra inhabiting the left half-plane that are universally realizable |
| title_full_unstemmed | Spectra inhabiting the left half-plane that are universally realizable |
| title_short | Spectra inhabiting the left half-plane that are universally realizable |
| title_sort | spectra inhabiting the left half plane that are universally realizable |
| topic | nonnegative matrix universal realizability jordan structure 15a18 15a29 |
| url | https://doi.org/10.1515/spma-2021-0155 |
| work_keys_str_mv | AT sotoricardol spectrainhabitingthelefthalfplanethatareuniversallyrealizable |