Minimal spectrally arbitrary sign patterns
An $n\times n$ sign pattern $\mathcal{A}$ is spectrally arbitrary if given any self-conjugate spectrum there exists a matrix realization of $\mathcal{A}$ with that spectrum. If replacing any nonzero entry of $\mathcal{A}$ by zero destroys this property, then $\mathcal{A}$ is a minimal spectrally arb...
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Veröffentlicht in: | SIAM journal on matrix analysis and applications 2004-01, Vol.26 (1), p.257-271 |
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Sprache: | eng |
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Zusammenfassung: | An $n\times n$ sign pattern $\mathcal{A}$ is spectrally arbitrary if given any self-conjugate spectrum there exists a matrix realization of $\mathcal{A}$ with that spectrum. If replacing any nonzero entry of $\mathcal{A}$ by zero destroys this property, then $\mathcal{A}$ is a minimal spectrally arbitrary sign pattern. Several families of sign patterns are presented that, for all $n\geq 3$, each contain an $n\times n$ minimal spectrally arbitrary sign pattern. These are the first families proven to have this property, and they improve previously known results. Furthermore, all $3\times 3$ minimal spectrally arbitrary sign patterns are determined, it is proved that any irreducible $n\times n$ spectrally arbitrary sign pattern must have at least $2n-1$ nonzero entries, and it is conjectured that the minimum number of nonzero entries is $2n$. |
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ISSN: | 0895-4798 1095-7162 |
DOI: | 10.1137/S0895479803432514 |