On the distance from a matrix to nilpotents
We prove that the distance from an \(n\times n\) complex matrix \(M\) to the set of nilpotents is at least \(\frac{1}{2}\sec\frac{\pi}{n+2}\) if there is a nonzero projection \(P\) such that \(PMP=M\) and \(M^*M\geq P\). In the particular case where \(M\) equals \(P\), this verifies a conjecture by...
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| Published in | arXiv.org |
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| Main Author | |
| Format | Paper |
| Language | English |
| Published |
Ithaca
Cornell University Library, arXiv.org
10.07.2023
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| Online Access | Get full text |
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| Summary: | We prove that the distance from an \(n\times n\) complex matrix \(M\) to the set of nilpotents is at least \(\frac{1}{2}\sec\frac{\pi}{n+2}\) if there is a nonzero projection \(P\) such that \(PMP=M\) and \(M^*M\geq P\). In the particular case where \(M\) equals \(P\), this verifies a conjecture by G.W. MacDonald in 1995. We also confirm a related conjecture in D.A. Herrero's book. |
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| ISSN: | 2331-8422 |